TITLE: Evaluating a triple summation ${\sum \sum \sum}_{1 \le i < j \le k \le 20} a_ia_ja_k$
QUESTION [4 upvotes]: If $$a_i= \begin{cases} 1 && 1 \le i \le 10 \\ 2 && 11 \le i \le 20 \end{cases}  $$
Find the value of $$\displaystyle {\sum \sum \sum}_{1 \le i < j \le k \le 20} a_ia_ja_k $$

I could only evaluate the summation when $1 \le i < j \le k \le 10$, by writing it as $\displaystyle {\sum _{i=1}^9\sum_{j=i+1}^{10} \sum_{k=j}^{10}} 1$ which came out to be $\displaystyle \binom{11}{3}$.
Should I make different cases like $i \le 10, 11 \le j \le k \le 20$ , etc.?
How do I evaluate it for the other cases? Is there any easier way to evaluate it without evaluating each sum one by one? Could there possibly be a combinatorial approach? I'd be delighted to see different approaches to this problem.

REPLY [2 votes]: Let

$ A = \sum_{i < j < k } a_ia_ja_k$
$ B = \sum_{i = j < k } a_ia_ja_k$
$ C = \sum_{i < j = k } a_ia_ja_k$
$ D = \sum_{i = j = k } a_ia_ja_k$

We're after $ A  + C$.
Clearly,

$ 30^3 = ( \sum a_i)^3  = D + 3C + 3B + 6A$.
$ D = 10 ( 1^3 + 2^3 ) = 90$
$C = 2^2 ( 10 \times 10\times 1 + \frac{10 \times 9 }{ 2} \times 2 ) + 1^2 (\frac{10 \times 9 } { 2} \times 1) = 805$
$ B = 1^2 (10\times 10 \times 2 + \frac{10 \times 9} { 2} \times 1) + 2^2 (\frac{10 \times 9 }{ 2} \times 2)  = 605$
So, $A = \frac{30^3 - 90 - 3 \times 805 - 3 \times 605 }{ 6} = 3780$.

Hence, $ A + C = 4585$.


TITLE: Relating Combinatorical Equations to Operations on Sets
QUESTION [0 upvotes]: In my university we learn Set Theory prior to starting Combinatorics but they don't seem to be making a clear and explicit connection between the two.  Yet it seems to me that there is in fact a very strong relation between well known combinatorial formulas like $D(n,k),c(n,k),p(n,k),n^k$ and the algebra of sets.  Could someone explain it and make it explicit?
Edit
Combinations with repetitions: $D(n,k)=\binom{n+k-1}{k}=\binom{n+k-1}{n-1} = \frac{(n+k-1)!}{(n-1)!}$
Combinations without reps: $c(n,k)=\binom{n}{k}=\frac{n!}{k!(n-k)!}$
Permutations with reps: $n^k$
Permutations without reps: $p(n,k)=\frac{n!}{(n-k)!}$

REPLY [1 votes]: Most combinatorial (families of) numbers count the elements in certain (families of) finite sets; this is the basis of enumerative combinatorics. Whether those families of finite sets play a very important role in set theory rather depends. Powers of $2$ count the powerset (collection of all subsets) of finite sets, which is an important notion in set theory. Combinations count those subsets with a fixed number of elements, which is somewhat less fundamental though still important in set theory. Powers of another number $n$ than $2$ count all maps to an $n$-element set, and $p(n,k)$ counts such maps that are injective; both these notions are fairly central to set theory. However combinations with repetitions $D(n,k)$ count multisets of $k$ elements chosen from an $n$ element set, which is a notion not usually encountered in set theory. They can be modeled by maps from an $n$-element set to $\mathbf N$ such that the sum of the values taken is $k$, but the importance of such a construction in set theory is not so obvious. On the other hand this number does occur in algebra as the number of monomials in $n$ (commuting) variables, of total degree $k$.


TITLE: Rationalise the Denominator
QUESTION [1 upvotes]: Rationalise the denominator and simplify fully:
$$\dfrac{6}{\sqrt{7} + 2}$$
I got the answer $\dfrac{2 \sqrt{7}}{3}$, but didn't get the mark. Is that not fully simplified?
I did $\displaystyle \frac{6}{\sqrt{7} + 2} \times \frac{\sqrt{7}}{\sqrt{7}} = \frac{6 \sqrt{7}}{(7 + 2)} = \frac{6 \sqrt{7}}{9} = \frac{2 \sqrt{7}}{ 3}$ .

REPLY [1 votes]: $$\frac{6}{\sqrt{7}+2}=\frac{6(2-\sqrt{7})}{(2+\sqrt{7})(2-\sqrt{7})} = \frac{12-6\sqrt{7}}{4-7} = \frac{3(4-2\sqrt{7})}{-3} = 2\sqrt{7}-4$$
Note that in general $$\frac{a}{b+c\sqrt{d}} = \frac{a(b-c\sqrt{d})}{(b+c\sqrt{d})(b-c\sqrt{d})} = \frac{ab-ac\sqrt{d}}{b^2-c^2d}$$
Notice that the denominator is now rational, i.e. in particular for this case it has no surds in it.

REPLY [1 votes]: I'll assume your expression was $\frac{6}{\sqrt{7}+2}$, since it's the only way you obtain that denominator.
$$\frac{6}{\sqrt{7}+2}\times\frac{\sqrt{7}-2}{\sqrt{7}-2}=\frac{6\sqrt{7}-12}{\sqrt{7}^2-2^2}=\frac{6\sqrt{7}-12}{3}={2\sqrt{7}-4}$$


TITLE: Hartshorne Proposition II.7.13a
QUESTION [3 upvotes]: Let $X$ be a Noetherian scheme, $\mathscr{I}$ be a coherent sheaf of ideals and let $\pi:\operatorname{Bl}_Y X \rightarrow X$ be the blow up of $X$ along the closed subscheme $Y$ corresponding to $\mathscr{I}$.
In the proof of the book of Hartshorne "Algebraic Geometry", Proposition II.7.13a, it is written:
$$\mathscr{S}(U)(1) = \bigoplus_{d \geq 0} \mathscr{I}^{d+1}(U)$$
where $\mathscr{S}$ is the sheaf of graded algebras defined by $\mathscr{S} = \bigoplus_{d \geq 0} \mathscr{I}^d$, and $U$ is an open affine subset of $X$. From the above equality it follows that $\mathscr{S}(U)(1) = \mathscr{I} \cdot \mathscr{S}(U)$, the ideal generated by $\mathscr{I}$ in $\mathscr{S}(U)$, and so the inverse image ideal sheaf $\pi^{-1}\mathscr{I} \cdot \mathcal{O}_X$ is equal to $\mathcal{O}_{\operatorname{Bl}_YX}(1)$.
My question is, why it is $\mathscr{S}(U)(1) = \bigoplus_{d \geq 0} \mathscr{I}^{d+1}(U)$? Shouldn't it be $\mathscr{S}(U)(1) = \bigoplus_{d \geq -1} \mathscr{I}^{d+1}(U)$, since $\mathscr{S}(U)(1)_{-1} = \mathscr{S}(U)_0 = \mathcal{O}_X(U) \neq 0$ ? In that case, is the equality $\mathscr{S}(U)(1) = \mathscr{I} \cdot \mathscr{S}(U)$ still true?

REPLY [1 votes]: First, recall that the functor which takes a graded $S$-module $M$ to the associated sheaf $\widetilde{M}$ on $\operatorname{Proj} S$ only cares about what happens in high-enough degree (see for instance exercise II.5.9(c)), so both Hartshorne's choice and your choice produce the same sheaf $\widetilde{\mathscr{I}}$ on the blowup. His choice has the advantage of making the next claim a little more obvious, though - using your choice, it's still true that the image of $\mathscr{S}(U)(1)$ in $\mathscr{S}(U)$ is the ideal generated by $\mathscr{I}$ in $\mathscr{S}(U)$, you just have the stuff in negative degree going to zero.


TITLE: Two fair coins, three unfair, one toss
QUESTION [1 upvotes]: some more questions about conditional probability:

You have five coins, of which two have heads on both sides, one has tails on both on both sides, and the other two are fair. You flip a randomly chosen coin. a) What is the probability that the lower side is heads? b-i) You open your eyes and see that the upper side is heads. What is the probability that the lower side is heads? b-ii) If you flip the coin again, what is the probability that the lower side is heads?

a) I'd say the phrasing here is a "trick", because in this particular case it doesn't matter if it's the upper or the lower side we're interested in. My answer is $\frac{2}{5} \cdot 1 + \frac{1}{5} \cdot 0 + \frac{2}{5} \frac{1}{2} = 0.6$.
b-i) This is asking for $P(\text{lower side is heads}| \text{upper side is heads})$. This equals $P(\text{lower side is heads AND upper side is heads})/P(\text{upper side is heads})$. The numerator is $2/5$ and the denominator $3/5$ by the argument in exercise (a), whence the answer is $2/3$.
So far so good?
b-ii) I'm a bit lost here because I'm really not sure how to interpret the question. I take it that the coin being tossed here is the one whose upper side previously came up as heads, and that we never actually inspected the lower side of that coin, whence we don't know what coin we're tossing now - well, we know it's the not the coin with tails on both sides, so is the answer $\frac{1}{2} \cdot 1 + \frac{1}{2} \frac{1}{2} = 3/4$?
Your hints and thoughts on all the exercises would be appreciated, because this all went way too easy :P
EDIT: This solution to (2b-ii) is due to user heropup (see comments): Let $A_1$ be the event that the two-heads coin was chosen, $A_2$ the fair coin, $A_3$ the two-tails coin, $L_i$ the event that the lower side is heads in the i:th coin flip, $U_i$ that the upper side is heads in the i:th coin flip.
To determine $P(L_2|U_1)$, put $Q(E) =P(E|U_1)$. Then by the law of total probability, $Q(L_2) = \sum_{i=1}^3 Q(A_i)Q(L_2|A_i)$, with the third summand clearly zero. The event $U_1$ doesn't impact $Q(L_2|A_i)$ for either $i$, whence $Q(L_2|A_i)=1/i$ and $Q(L2)= Q(A_1)+(1/2)Q(A_2)$. Finally, apply Bayes' thm to $Q(A_i)=P(A_i|U_1)$ to yield $P(L_2|U_1)=5/6$.

REPLY [2 votes]: Your answers to (a) and (b)(i) are correct.
To understand the last part, we are told that we flip the coin again without looking at the lower side.  We are also NOT told what the outcome of the second flip is.
The answer to (b)(i) gives us a mechanism by which to answer (b)(ii).  For an equivalent interpretation of the previous part is that the probability the coin we selected is two-headed is $2/3$, and the probability it is fair is $1/3$.  So the probability the next coin flip has the lower side heads is $(2/3)(1) + (1/3)(1/2) = 5/6$.


text:
\begin{document}
\thispagestyle{empty}
\begin{center}

\vspace{1.8cm} {\bf NON COMMUTATIVE SCALAR FIELDS FROM SYMPLECTIC DEFORMATION}\\
\vspace{1.5cm} {\bf M. Daoud}$^a${\footnote {\it Facult\'e des
Sciences, D\'epartement de Physique,
Agadir, Morocco; email: m$_-$daoud@hotmail.com}} and {\bf A. Hamama}$^b$
\\\vspace{0.5cm}
$^a$ {\it Abdus Salam International Centre for Theoretical Physics,\\
Strada Costiera 11, 34014 Trieste, Italy}\\ \vspace{0.2cm} $^b${\it
High Energy Laboratory, Faculty of Sciences, University Mohamed V,\\
P.O. Box 1014 , Rabat ,
Morocco}\\[1em]

\vspace{3cm} {\bf Abstract}
\end{center}
\baselineskip=18pt
\medskip
This paper is concerned with the quantum theory of noncommutative
scalar fields in two dimensional space time. It is shown that the
noncommutativity originates from the the deformation of symplectic
structures. The quantization is performed and the modes expansions of the fields,
in presence of an electro-magnetic background, are derived. The
Hamiltonian of the theory is given and the degeneracies lifting,
induced by the deformation, is also discussed.

\newpage
\section{Introduction}
In the last decade the noncommutative field theories have been
extensively investigated (for a review see [1-2]) in connection with
the low energy description of string theory with a constant magnetic
background in the presence of $D$-branes [3]. It is important to
note the construction of field theories in noncommutative space-time
encounters many problems as the violation  of Lorentz invariance,
the ultraviolet divergences, and the breaking of unitarity and
causality. However, despite such technical problems, the
noncommutative geometry still considered as one promising candidate
to provide a rigorous framework to investigate and to understand the
non
commutative space time structure at Planck scale [4] in relation with  quantum gravity.\\
It is also important to mention that  motivated by the
noncommutative field theories the quantum mechanical problems in
noncommutative space have received a considerable attention [5-9].
More recently it was shown that the noncommutative geometry via the
Moyal star product is relevant in many branches of physics  and
especially in condensed matter as for instance quantum Hall effect
in different geometries [10-11], bosonization in higher dimensions
[12] and braiding quantum statistics [13-19].\\

In this work, we shall be concerned with the quantum theory of
noncommutative scalar fields where the fields and their conjugate
momenta obey deformed equal time commutations rules. It is important to stress that the quantum theory of
noncommutative fields is defined on a commutative space-time
contrarily  to  the noncommutative field theories which arise from
the non-commutativity of space-time.  The interest for this new kind of quantum
theory of noncommutative fields, introduced in [20-24], is
essentially motivated by Lorentz invariance violation, the problem
of neutrino oscillations and the asymmetry of the dispersion
relation for particles and
antiparticles. \\

In this paper we propose a noncommutative formulation of two
dimensional scalar field theory underlying the noncommutative
algebra with a constant commutator of fields and simultaneously with
a constant commutator of momenta as well. This formulation is done
from a symplectic point of view. Indeed we first suggest a modification of the
symplectic structure associated to phase space associated
 to two massless scalar fields. Secondly, performing a dressing transformation, we
write the deformed two-form in a canonical form. The Hamiltonian is
then converted in a new one involving terms arising from the
deformation. We show that the dynamic described by a
deformed Hamiltonian and canonical two-form is equivalent to the description
which uses undeformed Hamiltonian and deformed symplectic structure. Note
that in a particular case, which will be specified below, the
construction developed here agrees with the results derived in [24]
and most importantly it start at classical level by deforming the
symplectic structure. In quantizing the theory the deformation
induces a modification of the commutation rules as well as in the
spectrum of the system. These results can be viewed as an extension of
the symplectic approach to non-commutative mechanics initiated in [25-27].  \\

The arrangement of this paper is as follows. In the section 2, we
first review some elements concerning symplectic structures in
deriving the evolution of classical fields and the canonical
quantization. This review, included here for completeness, is useful
and necessary to understand and to perform the approach developed in
the next sections. The section 3 concerns the deformation from a
symplectic point of view. We consider two classical scalar fields
and we modify the symplectic two-form to take into account the
presence of an electro-magnetic background. We define the
corresponding Poisson brackets from which we get ones between fields
and their conjugate momenta. In section 4, following the procedure
suggested in [20-24], we quantize the theory,  we propose the
Hamiltonian associated to a quantum theory of two noncommutative
scalar fields and we discuss the degeneracies lifting induced by the
deformation. Solving the Heisenberg equations of motion, we give the
modes expansion of the noncommutative fields. As mentioned above,
the approach essentially originates from the deformation of the
symplectic structures (symplectic two-form and Poisson brackets) in
the phase space of the scalar fields. We end section 4 with  some comments
concerning the  dressing transformation, which play an important
role in the
quantizing the model. Concluding remarks close this paper.\\

\section{Symplectic structures and field quantization}
We shall consider two real massless bosonic fields denoted by
$\Phi^i({\bf x})\equiv \Phi( x , t)$, $i = 1, 2$. The corresponding
action is
\begin{eqnarray}
S = \int_{\Sigma}dtdx {\cal L} =\frac{1}{2l}\sum_{i=1,2}
\int_{\Sigma}dtdx((\partial_t\Phi^i)(\partial_t\Phi^i) -
(\partial_x\Phi^i)(\partial_x\Phi^i)).
\end{eqnarray}
The space-time  region $\Sigma$ will be considered to be of the
form $[0 , l]\times[0, T]$ where $[ 0 , l]$ is line segment is the
spatial region. The equation of motion are given by the
variational principle such that the classical orbit of the fields
which connects the initial and final configurations $\Phi^i( x , 0)$ and $\Phi^i( x , T)$ at time $ t = 0$ and $ t = T$,
minimizes the action. Hence, the minimization condition $\delta S
= 0$ gives the equations of motion (Euler-Lagrange equations)
since the surface contribution of $\delta S$:
\begin{eqnarray}
\delta S_{surface} = \sum_{i=1,2} \int dtdx
\frac{\partial}{\partial x^{\mu}}\bigg(\frac{\partial{\cal
L}}{\partial(\partial_{\mu}\Phi^i)}\delta\Phi^i \bigg)
\end{eqnarray}
vanishes when we fix the initial and final fields configurations,
i.e. $\delta \Phi^i = 0$ at time $t = 0$ and $ t = T$ and also
assume that either $\delta \Phi^i = 0$ or $\frac{\partial {\cal
L}}{\partial(\partial_1\Phi^i)}$ vanishes at $x =0$ and $ x = l$ .
In equation (2) $\mu = 0 ,1$ stands for time and space directions
respectively. The summation over repeated indices is understood.
This is the standard way to get the trajectory of classical fields
by means of Euler-Lagrange equations. There exists another way
based on the symplectic structure of the phase space to describe
the classical evolution of the fields. Indeed, if one consider the
variations of the fields with $\delta \Phi^i$ non-vanishing at
time $t = 0$ and $t = T$, the integration of the surface
contribution term gives \\
\begin{eqnarray}
l\int dtdx \frac{\partial}{\partial
x^{\mu}}\bigg(\frac{\partial{\cal
L}}{\partial(\partial_{\mu}\Phi^i)}\delta\Phi^i \bigg) = {\cal
A}(T) - {\cal A}(0)
\end{eqnarray}
where
\begin{eqnarray}
{\cal A}(t) = \sum_{i=1,2}\int dx
\partial_{t}\Phi^i\delta\Phi^i.
\end{eqnarray}
This is the canonical one form, i.e. a differential one form on
the phase space. In equation (4),  $\delta $ denotes the exterior derivative
on the field space and the time derivatives of the fields must be
treated as variables since ${\cal A}$ is defined at fixed time.
The phase space whose variables (coordinates) are $\Phi^i(x,t)$
and the canonical momentum $\Pi^i(x,t) =
\partial_{t}\Phi^i(x,t)$ is now  equipped  with a
symplectic structure, a canonical two-form, defined by
\begin{eqnarray}
\Omega_0 = \delta{\cal A}(t) = \sum_{i=1,2}\int dx
\delta\Phi^i\wedge\delta\Pi^i
\end{eqnarray}
which can be seen as field strength  corresponding to one-form
${\cal A}$ viewed as $U(1)$ gauge potential. It can also be
written as
\begin{eqnarray}
\Omega_0 = \frac{1}{2}\sum_{I,J}\int dx dx'(\Omega_0)_{IJ}(x,x')
\delta\xi^{I}(x)\wedge\delta\xi^{J}(x')
\end{eqnarray}
where we denote the phase space coordinates by $\xi^I(x) \equiv
\xi^{ii'}(x) = \Phi^i(x)$ (resp. $\Pi^i(x)$) for $i'=1$ (resp.
for $i'=2$) and
\begin{eqnarray}
(\Omega_0)_{IJ}(x,x')\equiv (\Omega_0)_{IJ}\delta(x-x')=
\delta_{ij}\epsilon_{i'j'}\delta(x-x').
\end{eqnarray}
It follows that the Poisson bracket of two functionals ${\cal F}$
and ${\cal G}$ is
\begin{eqnarray}
\{{\cal F} , {\cal G}\} = \sum_{I,J}\int dx
(\Omega_0)^{IJ}\frac{\delta {\cal F}}{\delta\xi^{I}}\frac{\delta
{\cal G}}{\delta\xi^{J}}
\end{eqnarray}
where $\Omega_0^{IJ}$ are the elements of the inverse matrix  of
$\Omega_{IJ}$. In particular the Poisson brackets for the
coordinates on the phase space are the inverse of the symplectic
two-form (treated as matrix):
\begin{eqnarray}
\{\xi^{ik}(x') , \xi^{jl}(x)\} =
\delta^{ij}\epsilon^{kl}\delta(x-x'),
\end{eqnarray}
or alternatively
\begin{eqnarray}
\{ \Phi^i( x , t) , \Phi^j( x' , t)\} = 0{\hskip 0.5cm} \{
\Phi^i( x , t) , \Pi^j( x',t)\} = \delta^{ij}\delta(
x- x'){\hskip 0.5cm}\{ \Pi^i( x , t) , \Pi^j(
x' ,t)\}= 0.
\end{eqnarray}
As we are concerned with two component scalar field theory in a
compact domain, with the boundary condition we can expand the
fields as
\begin{eqnarray}
\Phi^i(  x , t ) = \frac{1}{\sqrt{ l}}\sum_{n \in \mathbf{Z}}
q_n^i\exp(-i n x)
\end{eqnarray}
 and
 \begin{eqnarray}
\Pi^i( x , t ) = \frac{1}{\sqrt{ l}}\sum_{n \in \mathbf{Z}}
p_n^i \exp(-i n x)
\end{eqnarray}
where the normal or Fourier modes $q_n$ and $p_n$ satisfy the
conditions
\begin{eqnarray}
 q_{-n} = q_n^{\star} {\hskip 1.5cm}p_{-n} = p_n^{\star}
\end{eqnarray}
 required by the reality of fields and  their time
 derivatives.\\
Using the equations (5) and (11-12), the symplectic two-form
$\Omega_0$ rewrites
\begin{eqnarray}
\Omega_0 =\sum_{i=1,2} \sum_{n\in {\bf Z} }\delta q^i_n \wedge
\delta p^i_{-n}
\end{eqnarray}
and the Poisson brackets take the simple form
\begin{eqnarray}
\{{\cal F} , {\cal G}\} = \sum_{i=1,2}\sum_{n\in {\bf Z}}
\frac{\delta {\cal F}}{\delta q^{i}_n}\frac{\delta {\cal G}}{\delta
p^{i}_{-n}} - \frac{\delta {\cal F}}{\delta p^{i}_{-n}}\frac{\delta
{\cal G}}{\delta q^{i}_n}.
\end{eqnarray}
In particular, the Poisson brackets corresponding to the canonical
coordinates of the phase space generated by the Fourier modes
$q_n^i$ and $p_n^i$ are given by
\begin{eqnarray}
\{ q_n^i , q_m^j\} = 0 {\hskip 0.5cm} \{ p_n^i , p_m^j\} = 0 {\hskip
0.5cm} \{  q_n^i, p_m^j\} = \delta_{i,j}\delta_{m+n,0}.
\end{eqnarray}
The lagrangian is given by
\begin{eqnarray}
L = \int dx {\cal L} = \frac{1}{2}\sum_{in} (p_n^i p_{-n}^i - n^2
q_n^iq_{-n}^i)
\end{eqnarray}
in terms of the Fourier modes. The canonical Hamiltonian reads
\begin{eqnarray}
H_0 = \sum_{i=1,2}\int dx \Pi^i\partial_t\Phi^i - \int dx {\cal L} =
\frac{1}{2}\sum_{in} (p_n^i p_{-n}^i + n^2 q_n^iq_{-n}^i).
\end{eqnarray}
Since we defined $p^i_n = {\dot q}^i_n$, the canonical momenta
conjugate to $q_n^i$ is defined by
\begin{eqnarray}
\pi_n^i = \frac{\partial L}{\partial\dot{q}_n^i} = \dot{q}_{-n}^i =
p^i_{-n}.
\end{eqnarray}
The Hamiltonian is the generator of time translation. The time
evolution of a function ${\cal F}$ is given by the Hamilton's
equation of motion $\dot{{\cal F}}=\{ {\cal F} , H_0\}$ that gives
\begin{eqnarray}
\frac{dq_n^i}{dt} = \{ q_n^i, H_0\} =  p_{n}^i {\hskip 1cm}
\frac{dp_{n}^i}{dt} = \{ p_{n}^i, H_0\} =-n^2q_{-n}^i
\end{eqnarray}
To pass over the quantum theory in the Heisenberg picture, all
canonical variables become Heisenberg operators satisfying
commutation relations corresponding to Poisson brackets as
\begin{center}
(Poisson Bracket) $\rightarrow$ $-i$ (commutator)
\end{center}
The equations (20) are easily solvable and one recover the
evolution of the fields which can also be derived from the
Euler-Lagrange equations. However, the symplectic prescription
discussed has the merit to be more appropriate to introduce
deformed target space in the spirit of the recent results obtained
in [20-23] and [24]. This issue is the purpose of the next
section.

\section{{Deformed symplectic structure}}

Hereafter, we consider two components abelian field theory in two
dimensional space-time. The generalization to arbitrary number of
components is straightforward. As mentioned above the dynamic of the
system is governed by the Hamiltonian $H_0$ (18). We now assume that
the symplectic structure of the phase space is modified due to the
interaction between the field components $\Phi^1$ and $\Phi^2$ in presence
of an  electromagnetic background. This can be formulated by replacing
the canonical two-form $\Omega_0$ by a closed new one as follows
\begin{eqnarray}
\Omega = \Omega_0 + \frac{1}{2} {\cal E}_{ij} \int dx
\delta\xi^{i2}(x) \wedge\delta\xi^{j2}(x)- \frac{1}{2}{\cal
B}_{ij}\int dx \delta\xi^{i1}(x)\wedge\delta\xi^{j1}(x)
\end{eqnarray}
where the deformation is encoded in the constant antisymmetric
tensors ${\cal E}_{ij}$ and ${\cal B}_{ij}$.\\
Alternatively, $\Omega$ given by (21) can be cast in a compact form
\begin{eqnarray}
\Omega = \frac{1}{2}  \int dxdx' \Omega_{IJ}(x,x') \delta\xi^{I}(x)
\wedge\delta\xi^{J}(x')
\end{eqnarray}
providing the non trivial deformed metric
\begin{eqnarray}
\Omega_{IJ}(x,x') = \Omega_{IJ}\delta (x-x')
\end{eqnarray}
with
\begin{eqnarray}
\Omega_{IJ} = \delta_{ij}\epsilon_{i'j'} + {\cal
E}_{ij}\delta_{i'2}\delta_{j'2} - {\cal
B}_{ij}\delta_{i'1}\delta_{j'1}
\end{eqnarray}
which is non degenerate (det$\Omega \neq 0$) when the antisymmetric
tensors ${\cal E}_{ij}$ and ${\cal B}_{ij}$ satisfy the condition
$\det( 1_{2\times 2} - {\cal E}{\cal B}) \neq 0$. This is easily
seen in writing $\Omega$ in a matrix form. We assume in this work
that such a condition is satisfied. To find the classical equations
of motion and to establish the connection between the classical and
quantum theory, it is necessary to define the Poisson brackets
associated with the new phase space geometry in a consistent way.
Indeed, recalling that the Poisson brackets for the coordinates on
the phase space are the inverse of the symplectic form as matrix, we
have
\begin{eqnarray}
\{{\cal F} , {\cal G}\} = \sum_{I,J}\int dx \Omega^{IJ}\frac{\delta
{\cal F}}{\delta\xi^{I}}\frac{\delta {\cal G}}{\delta\xi^{J}}
\end{eqnarray}
where $\Omega^{IJ}$ is the inverse matrix of $\Omega_{IJ}$ (24) and
${\cal F}$ and ${\cal G}$ are two functionals on the phase space. \\
As we are interested by scalar field theory, we use the
equations (11-12) and (22) to write the symplectic form  as
\begin{eqnarray}
\Omega = \delta q^{i}_{n}\wedge\delta p^{i}_{-n}+\frac{1}{2}{\cal
E}_{ij}\delta p^{i}_{n}\wedge\delta p^{j}_{-n}- \frac{1}{2}{\cal
B}_{ij}\delta q^{i}_{n}\wedge\delta q^{j}_{-n}
\end{eqnarray}
and the
Poisson brackets take the simple form
\begin{eqnarray}
\{{\cal F}, {\cal G}\} = \sum_{ikn} (\omega^{-1}_1)_{ik}\frac{\delta
{\cal F}}{\delta q^i_n}\bigg[ \frac{\delta {\cal G}}{\delta
p^k_{-n}} - \sum_{j} {\cal E}_{kj}\frac{\delta {\cal G}}{\delta
q^j_{-n}} \bigg] -  (\omega^{-1}_2)_{ik}  \frac{\delta {\cal
F}}{\delta p^i_{-n}}\bigg[ \frac{\delta {\cal G}}{\delta q^k_n}-
\sum_j{\cal B}_{kj}\frac{\delta {\cal G}}{\delta p^j_n}\bigg]
\end{eqnarray}
where the functionals ${\cal F}$ and ${\cal G}$ are now expressed in
terms of $q^i_n$ and $p^i_n$ generating the phase space. The
matrix elements of $\omega_1$ and $\omega_2$, occurring in (27), are defined by
\begin{eqnarray}
(\omega_1)_{ij} = \delta_{ij} - {\cal E}_{ik}{\cal B}_{kj},
\end{eqnarray}
\begin{eqnarray}
(\omega_2)_{ij} = \delta_{ij} - {\cal B}_{ik}{\cal E}_{kj}.
\end{eqnarray}
The last equations can be read in matrices form as $\omega_1 = 1 -
{\cal E}{\cal B}$ and $\omega_2 = 1 - {\cal B}{\cal E}$,
respectively. It follows that the modified canonical Poisson
brackets are
\begin{eqnarray}
\{ q^i_n , q^j_m\} = - \sum_k(\omega^{-1}_1)_{ik}{\cal
E}_{kj}\delta_{m+n,0},
\end{eqnarray}
\begin{eqnarray}
\{ p^i_n , p^j_m\} =  \sum_k(\omega^{-1}_2)_{ik}{\cal
B}_{kj}\delta_{m+n,0},
\end{eqnarray}
\begin{eqnarray}
\{ q^i_n , p^j_{-m}\} =  (\omega^{-1}_1)_{ij}\delta_{n,m} =
(\omega^{-1}_2)_{ji}\delta_{n,m}.
\end{eqnarray}
According to the modification of the symplectic structure of the phase
space, we introduce the vector fields $X_{\cal F}$ associated to a
given functional  ${\cal F}(q^i_n , p^i_n)$
\begin{eqnarray}
X_{\cal F} = \sum_{in} X^i_n \frac{\delta }{\delta q^i_n} +
Y^i_n\frac{\delta }{\delta p^i_{-n}}
\end{eqnarray}
such that the interior contraction of $\Omega$ with  $X_{\cal F}$
gives
\begin{eqnarray}
{\it i}(X_{\cal F}) \Omega = \delta {\cal F}.
\end{eqnarray}
A straightforward calculation leads to
\begin{eqnarray}
 X^i_n= \sum_j(\omega^{-1}_1)_{ij}\bigg(\frac{\delta {\cal
F}}{\delta p^j_{-n}} - \sum_k{\cal E}_{jk} \frac{\delta {\cal F}
}{\partial q^k_{-n}}\bigg)
\end{eqnarray}
\begin{eqnarray}
 Y^i_n=- \sum_j(\omega^{-1}_2)_{ij}\bigg(\frac{\delta {\cal
F}}{\delta q^j_{n}} - \sum_k{\cal B}_{jk} \frac{\delta {\cal F}
}{\partial p^k_{n}}\bigg)
\end{eqnarray}
 and one can check that
\begin{eqnarray}
{\it i}(X_{\cal F}){\it i}(X_{\cal G}) \Omega = \{{\cal F}, {\cal
G}\}.
\end{eqnarray}
Thus in the deformed case the fields and their conjugate momentum
satisfy the following Poisson algebra
\begin{eqnarray}
\{ \Phi^i( x , t) , \Phi^j( x',t)\} = -
\sum_k(\omega^{-1}_1)_{ik}{\cal E}_{kj}\delta(x - x'),
\end{eqnarray}
\begin{eqnarray}
 \{ \Phi^i( x ,
t) , \Pi^j( x' ,t)\} =  (\omega^{-1}_2)_{ij}\delta( x -
x'),
\end{eqnarray}
\begin{eqnarray}
\{ \Pi^i( x , t) , \Pi^j(  x' ,t)\}=
\sum_k(\omega^{-1}_2)_{ik}{\cal B}_{kj}\delta( x - x').
\end{eqnarray}
Clearly, in the limiting case ${\cal E} = 0$ and ${\cal B} = 0$, we
recover the canonical Poisson brackets (10).\\
To simply our purpose, let us now set
\begin{eqnarray}
{\cal E}_{ij} = \theta {\epsilon}_{ij} {\hskip 1.5cm}{\cal B}_{ij} =
\bar{\theta}{\epsilon}_{ij}
\end{eqnarray}
where $\epsilon_{ij}$ is the usual antisymmetric tensor
($\epsilon_{12} = - \epsilon_{21} = 1$). With this choice, the
Poisson brackets (30-32) read simply
\begin{eqnarray}
\{ q^i_n , q^j_m\} = -
\frac{\theta}{1+\theta\bar{\theta}}\epsilon_{ij}\delta_{m+n,0}
\end{eqnarray}
\begin{eqnarray}
\{ p^i_n , p^j_m\} =
\frac{\bar{\theta}}{1+\theta\bar{\theta }}\epsilon_{ij}\delta_{m+n,0}
\end{eqnarray}
\begin{eqnarray}
\{ q^i_n , p^j_{-m}\} =
\frac{1}{1+\theta\bar{\theta }}\delta_{ij}\delta_{n,m}
\end{eqnarray}
reflecting a deviation from the canonical brackets. At this stage,
it is remarkable that the symplectic form (26) and  the
corresponding Poisson brackets (27) can be converted in the canonical
forms by mean of the so-called dressing transformation  which
furnishes  a simply way to quantize the theory. This is the main
task of the next section.
\section{ Dressing transformation and Quantization}
To begin note that under the following transformation
\begin{eqnarray}
Q^i_n = a q^i_n + \frac{1}{2} b\theta\sum_k\epsilon_{ki} p^k_{n}
\end{eqnarray}
\begin{eqnarray}
P^i_n = c p^i_n + \frac{1}{2} d\bar{\theta}\sum_k\epsilon_{ki}
q^k_{n},
\end{eqnarray}
the Poisson brackets (42-44) give the canonical ones
\begin{eqnarray}
\{ Q^i_n , Q^j_m\} =  0  {\hskip 1.5cm } \{ P^i_n , P^j_m\} = 0
{\hskip 1.5cm } \{ Q^i_n , P^j_{-m}\} = \delta_{ij}\delta_{n,m}
\end{eqnarray}
when the scalars $a$, $b$, $c$ and $d$ satisfy the following
constraints
$$
4a^2 - 4ab-\theta\bar{\theta}b^2 = 0
$$
$$
4c^2 - 4cd-\theta\bar{\theta}d^2 = 0
$$
$$
4ac + 2\theta\bar{\theta}(ad+bc)-\theta\bar{\theta}bd = 4(1 +
\theta\bar{\theta}).
$$
The transformation (45-46) generalizes the so-called dressing transformation introduced in [24].
As  simple solution of the above equations, we consider
\begin{eqnarray}
a = c = \frac{1}{b} = \frac{1}{d}=
\frac{1}{\sqrt{2}}\sqrt{1+\sqrt{1+\theta\bar{\theta}}}.
\end{eqnarray}
With these new dynamical variables, $\Omega$ takes a canonical form
\begin{eqnarray}
\Omega = \sum_{in} \delta Q^i_n \wedge \delta P^i_{-n}.
\end{eqnarray}
Inverting the above dressing transformation (45-46)
\begin{eqnarray}
q^i_n = \frac{a}{\sqrt{1+\theta{\bar\theta}}}\bigg[ Q^i_n +
\frac{\theta}{2a^2} \sum_k\epsilon_{ik} P^k_{n}\bigg]
\end{eqnarray}
\begin{eqnarray}
p^i_n = \frac{a}{\sqrt{1+\theta{\bar\theta}}}\bigg[P^i_n +
\frac{\bar{\theta}}{2a^2}\sum_k\epsilon_{ik} Q^k_{n}\bigg],
\end{eqnarray}
the Hamiltonian (18), denoted in what follows by $H$ to avoid any
confusion, becomes
\begin{eqnarray}
H = \frac{a^2}{2(1+\theta{\bar\theta)}}\bigg[ \sum_{in} (1 +
\frac{\theta^2}{4a^4}n^2) P^i_nP^i_{-n} +
(\frac{\bar{\theta}^2}{4a^4} + n^2)Q^i_nQ^i_{-n} +
(\frac{\theta}{a^2}n^2 + \frac{\bar{\theta}}{a^2})
\sum_j\epsilon_{ij}P^i_{-n}Q^j_n\bigg].
\end{eqnarray}
It is important to note that thanks to the dressing transformation
(45-46), the dynamic described by the Hamiltonian $H_0$ on the
deformed symplectic space is converted to one described by
underformed (canonical) symplectic structure (49) and the Hamiltonian
(52) expressed in terms of the new dynamical modes of the theory.
Evidently the ($\theta ,\bar{\theta}$)-dependent terms in $H$ arise
from the deformation of the symplectic structure. This suggests that
the fields $\Phi^1$ and $\Phi^2$ interacts with a given potential $
V_{int} $
 to have $H = H_0 + V_{int}$ where $H_0$ is the free Hamiltonian
given by (18) modulo the substitution $(q,p) \rightarrow (Q,P) $. In
this respect the deformation of the symplectic structure  can be
seen as a perturbation reflecting the action of  some external
potential on the system. This feature is very similar to the Landau problem in quantum mechanics.\\

We now come to the quantization of the model. Since, as mentioned
early, the two descriptions are equivalent, we shall quantize the
system using the canonical prescription. Thus, we replace the phase
space variables by operators satisfying commutation rules where the
commutators are given by $i$ times the associated Poisson brackets.
Consequently, we have
\begin{eqnarray}
[Q^i_n , Q^j_m] = 0 {\hskip 1cm}[P^i_n , P^j_m]= 0 {\hskip
1cm}[Q^i_n , P^j_{-m}]= i\delta_{m+n,0}\delta_{i,j}.
\end{eqnarray}
This gives
\begin{eqnarray}
[q^i_n , q^j_m]=
-i\frac{\theta}{1+\theta\bar{\theta}}\epsilon_{ij}\delta_{m+n,0}
\end{eqnarray}
\begin{eqnarray}
 [p^i_n , p^j_m] = i\frac{\bar{\theta}}{1+\theta\bar{\theta}}\epsilon_{ij}\delta_{m+n,0}
\end{eqnarray}
\begin{eqnarray}
[q^i_n , p^j_{-m}]=
i\frac{1}{1+\theta\bar{\theta}}\delta_{i,j}\delta_{n,m},
\end{eqnarray}
from which we obtain the equal time commutation relations for the
fields $\Phi^1$, $\Phi^2$ and their conjugate momenta:
\begin{eqnarray}
[\Phi^i( x , t) , \Phi^j( x',t) ] = -i
\frac{\theta}{1+\theta\bar{\theta}}\epsilon_{ij} \delta(
x- x')
\end{eqnarray}
\begin{eqnarray}
[\Pi^i( x ,t) , \Pi^j( x' , t) ] = i
\frac{\bar{\theta}}{1+\theta\bar{\theta}}\epsilon_{ij}
\delta( x -  x')
\end{eqnarray}
\begin{eqnarray}
[ \Phi^i( x ,t ), \Pi^j( x' , t)] = - i
\frac{1}{1+\theta\bar{\theta}}\delta_{ij}\delta( x -
x').
\end{eqnarray}
At this stage, we introduce the operators
\begin{eqnarray}
a^i_n = \sqrt{\frac{\Delta_n}{2}} (Q^i_n + i
\frac{P^i_{n}}{\Delta_n}) {\hskip 1cm} a^{i+}_n =
\sqrt{\frac{\Delta_n}{2}} (Q^i_{-n} - i \frac{P^i_{-n}}{\Delta_n})
\end{eqnarray}
where
\begin{eqnarray}
\Delta_n = \sqrt{\frac{\bar{\theta}^2 + 4a^4n^2}{4a^4 +
\theta^2n^2}}.
\end{eqnarray}
They satisfy the usual Heisenberg commutation relations $[a^i_n ,
a^j_m] = \delta_{i,j}\delta_{m,n}$. The Hamiltonian $H$ can be
written as the sum of two contributions:
\begin{eqnarray}
H-H_L = \frac{1}{4}\frac{1}{1+\theta\bar{\theta}}\sum_{in}
\sqrt{(\bar{\theta}^2 + 4a^4
n^2)(4a^4+\theta^2n^2)}(1+2a^{i+}_na^i_n)
\end{eqnarray}
and
\begin{eqnarray}
H_L = \frac{i}{2}\frac{1}{1+\theta\bar{\theta}}\sum_{ijn}
(\bar{\theta} + \theta n^2) \epsilon_{ij}a^{i+}_na^j_n.
\end{eqnarray}
The Hamiltonian $H$ can be diagonalized by considering the operators
\begin{eqnarray}
A^1_n = \frac{1}{\sqrt{2}}(a^1_n -i a^2_n) {\hskip 1cm}A^2_n =
\frac{1}{\sqrt{2}}(a^1_n +i a^2_n).
\end{eqnarray}
Indeed, substituting (64) in (62-63), we obtain
\begin{eqnarray}
H = \sum_n  [\omega_n + (\omega_n - \bar{\omega}_n)A^{1+}_nA^1_n  +
(\omega_n + \bar{\omega}_n)A^{2+}_nA^2_n ]
\end{eqnarray}
where
\begin{eqnarray}
\omega_n = \frac{1}{2}\frac{1}{1+\theta\bar{\theta}}
\sqrt{(\bar{\theta}^2 + 4a^4 n^2)(4a^4+\theta^2n^2)}
\end{eqnarray}
and
\begin{eqnarray}
\bar{\omega}_n = \frac{1}{2(1+\theta\bar{\theta})}
(\bar{\theta} + \theta n^2)
\end{eqnarray}
From the last three equations, it is easily seen that the
deformation induces a lifting of the degeneracies of the spectrum.
The Hamiltonian is a superposition of two independents one
dimensional oscillator unlike to undeformed case where the
Hamiltonian is a sum of two-dimensional oscillators. The dynamics of
this system is described by the Heisenberg equations:
\begin{eqnarray}
\frac{dA^1_n}{dt} = -i[ A^1_n , H ] = -i \omega_n^- A_n^1 {\hskip
1cm}\frac{dA^2_n}{dt} = -i[ A^2_n , H ] = -i \omega_n^+ A_n^2
\end{eqnarray}
where $\omega^{\pm}_n = \omega_n \pm {\bar{\omega}}_n $. Thus, we
have:
\begin{eqnarray}
A^1_n(t) = \hat{A}_n^1\exp(-i \omega_n^-t)  {\hskip 1cm}A^1_n(t) =
\hat{A}_n^2\exp(-i \omega_n^+t)
\end{eqnarray}
where the operators $\hat{A}^1_n$ and $\hat{A}^2_n$ are
time-independents. Consequently, using the equations (50), (60), (64) and
(69) we obtain the normal modes of the model as
\begin{eqnarray}
q^1_n(t) = \frac{1}{2}\bigg[\Lambda_n^+ \bigg(
\hat{A}^1_{n}\exp(-i\omega_n^-t) +
\hat{A}^{1+}_{-n}\exp(+i\omega^-_nt)\bigg) + \Lambda_n^- \bigg(
\hat{A}^2_{n}\exp(-i\omega_n^+t) +
\hat{A}^{2+}_{-n}\exp(+i\omega_n^+t)\bigg)\bigg]
\end{eqnarray}
 and
\begin{eqnarray}
q^2_n(t) = \frac{i}{2}\bigg[\Lambda_n^+ \bigg(
\hat{A}^1_{n}\exp(-i\omega_n^-t) -
\hat{A}^{1+}_{-n}\exp(+i\omega^-_nt)\bigg) + \Lambda_n^- \bigg(
\hat{A}^{2+}_{-n}\exp(+i\omega_n^+t)-\hat{A}^2_{n}\exp(-i\omega_n^+t)\bigg)\bigg]
\end{eqnarray}
where the $(\theta , \bar{\theta})$-dependent constants
$\Lambda_n^{\pm}$ are defined by
\begin{eqnarray}
\Lambda_n^{\pm} = \bigg[\frac{1}{\sqrt{\Delta_n}} \pm
\frac{\theta}{2a^2}\sqrt{\Delta_n}\bigg].
\end{eqnarray}
It is clear that in the limiting case $\theta = 0$ and $\bar{\theta}
= 0$, one recovers the usual results arising from the equations
(20). Note also that for $\bar{\theta} = 0$, our results agree with ones obtained in [24].
 We end up this section by some remarks related  the dressing transformation (45-46).
 In this sense, we will  show that the deformed Poisson algebra Eqs.(42-44) is
un-equivalent to un-deformed one (47). The un-equivalency occurs if
the two algebras can not be transformed to each other by a unitary
transformation. This means that the dressing transformation (45-46)
should be not unitary or more precisely not orthogonal since its
elements are reals. In fact , it is easy to check that the
orthogonality  requires the following conditions
\begin{eqnarray}
a^2 + \frac{1}{4}\theta^2 b^2 = 1 {\hskip 1cm} c^2 +
\frac{1}{4}{\bar{\theta}}^2 d^2 = 1  {\hskip 1cm} bc \theta = ad
\bar{\theta}.
\end{eqnarray}
Setting
\begin{eqnarray}
 a = \cos\varphi_1{\hskip 0.5cm} c =  \cos\varphi_2    {\hskip 0.5cm} b\theta = 2  \sin\varphi_1 {\hskip 0.5cm}
 d\bar{\theta} = 2  \sin\varphi_2
\end{eqnarray}
one can see that last equality in (73) is satisfied if
\begin{eqnarray}
  \varphi_1 = \varphi_2 + n\pi {\hskip 1cm} n\in \textbf{N}
\end{eqnarray}
which implies
\begin{eqnarray}
a = \pm c {\hskip 1cm} b\theta = \pm d \bar{\theta}
\end{eqnarray}
Consistency with (48) gives $\theta = \bar{\theta}$. This shows that
the dressing transformation is orthogonal when the strengths of
magnetic and electric backgrounds are equal. Finally, we stress that the transformation
(45-46) is similar to Darboux coordinates transformation (see for instance [26-27]).\\

\section{Concluding remarks }
We have clarified a procedure generating the noncommutative scalar
field theories. The key point of this procedure is the deformation
of the symplectic structure of the phase space of classical fields.
Having constructed the Poisson brackets, we quantized the model
under consideration following the standard canonical scheme thanks
to the so called dressing transformation(45-46).\\
An interesting feature of this approach lies on the deformation of
the symplectic structure which is introduced before the quantization
process.  The present results can be extended in various directions.
In particular, we believe that this approach can be adapted to the
theory of noncommutative chiral fields in two dimension in relation
with fractional quantum Hall effect. Indeed, it is well established
that for an incompressible quantum Hall droplet, the edges
excitations are described by a chiral scalar field. In this respect,
the quantum theory of noncommutative chiral fields
can provides us with an unified scheme to classify different Hall hierarchies and
 can bring new fractional filling factors with interesting physical consequences. We hope to report on this issue in a
forthcoming work.

{\vskip 1cm}
 \noindent {\bf Acknowledgments}\\
MD would like to thank the hospitality and kindness of Condensed
 Matter and Statistical Physics section of Abdus Salam International
 Centre for Theoretical Physics (AS-ICTP) where this work was
 done. The authors are indebted to the referee for his
 constructive comment.

config: arxiv


text:
{\bf Problem.} All the positive integers greater than 1 are arranged in five columns (A, B, C, D, E) as shown. Continuing the pattern, in what column will the integer 800 be written?

[asy]
label("A",(0,0),N);
label("B",(10,0),N);
label("C",(20,0),N);
label("D",(30,0),N);
label("E",(40,0),N);
label("Row 1",(-10,-7),W);
label("2",(10,-12),N);
label("3",(20,-12),N);
label("4",(30,-12),N);
label("5",(40,-12),N);
label("Row 2",(-10,-24),W);
label("9",(0,-29),N);
label("8",(10,-29),N);
label("7",(20,-29),N);
label("6",(30,-29),N);
label("Row 3",(-10,-41),W);
label("10",(10,-46),N);
label("11",(20,-46),N);
label("12",(30,-46),N);
label("13",(40,-46),N);
label("Row 4",(-10,-58),W);
label("17",(0,-63),N);
label("16",(10,-63),N);
label("15",(20,-63),N);
label("14",(30,-63),N);
label("Row 5",(-10,-75),W);
label("18",(10,-80),N);
label("19",(20,-80),N);
label("20",(30,-80),N);
label("21",(40,-80),N);
[/asy]
{\bf Level.} Level 2
{\bf Type.} Counting & Probability
{\bf Solution.} We write the sequence of columns in which the integers are written: \[B,C,D,E,D,C,B,A,B,C,D,E,\ldots.\] We see that the sequence consists of the block \[B,C,D,E,D,C,B,A\] repeated over and over.

Now, note that 800 will be the $799^{\text{th}}$ number in the list $\allowbreak \{2,3,4,\ldots,\}$. When 799 is divided by 8, its remainder is 7, so 800 will be written in the same column in which the seventh number is written. This is column B.

As another solution, we could note that all multiples of 8 appear in column B, so 800 must be in column $\boxed{\text{B}}$.

set_name: MATH


TITLE: Dynamic field without kinetic term
QUESTION [7 upvotes]: I am trying to read a paper (https://arxiv.org/abs/2206.07725) that models electric and magnetic charges on a lattice model. But I am new to lattice theories, so I might be misunderstanding a lot of things here. I do have a basic understanding of QFT. My confusion stemmed while reading this paper, but I think my question is general enough. So it might be answerable without reference to this paper. But let me describe it for the sake of completeness. In this model, the authors have defined charge current $n_\mu$ on the links ( which is coupled to the gauge field $a_\mu$):
$$ S  = \frac{1}{4g^2} \sum_P F^2_{\mu\nu} + \frac{iN\theta}{16 \pi^2} \sum_{r,R} F_{\mu\nu}(r)K(r-R)\tilde{F}_{\mu\nu}(R) - iN \sum_l n_\mu a_\mu \hspace{2cm}  (3.2)$$
( Here l are links, and P are plaquettes ). So basically, we are defining the electric charge current $n_\mu$ running over the links. To compute the partition function Z, we are summing over all possible configurations of $n_\mu$.
Note that in this action, there is NO kinetic term for the charged matter carrying electric current. Yet we are summing over all $n_\mu$ configurations (which is akin to doing path integral over the charge carrying field). In the third last line on page 10, authors say:

"This lattice gauge theory contains both $\textbf{dynamical}$ charges and monopoles."

From this, what I understand is that the matter field ( that leads to the current $n_\mu$ ) is not a background field. It is dynamic and hence fluctuating.
But I am still confused. So my confusion is:

Is it okay to path integrate a field without having its kinetic term?
Why is there no kinetic term?
What would it represent physically? I mean, do we have (charged) matter field in the theory or not.

My best guess is that such a theory will indeed represent "dynamic" electric charges i.e. they can be pair produced. But such  "electron/positron" particles will NOT propagate since there is no kinetic term for this field.
I might have said many incorrect things. Please feel free to educate me on obvious things as well.

REPLY [5 votes]: I will answer your questions in their field-theoretic form, in a general setting, and then I will explain in what guise they appear, in the referenced paper [1]. $\newcommand{\d}{\mathrm{d}}$
Tl;dr:

It is okay to integrate a field without kinetic terms, these usually represent Lagrange multiplier or discrete gauge fields. The case in [1] is a Lagrange multiplier forcing a gauge field to be discrete.
The fields in question in [1] have no kinetic terms because they are not allowed by symmetry, since they are conserved currents.
Finally, the charged matter has been integrated out. The conserved current term is the most relevant term in the EFT of the matter, and integrating over it completes integrating out the matter.

Is it okay to integrate a field without kinetic terms?
There are various occasions in which you need to path integrate over a field that has no kinetic terms.

A very common occasion is a Lagrange multiplier. Say you have a field theory, with dynamical fields $\phi$, and you want to enforce the constraint $\mathcal{C}(\phi)$. In the path integral you would write something like
$$ Z = \int \mathrm{D}\phi\ \delta[\mathcal{C}(\phi)] \exp(-S[\phi])$$
and then you could represent the delta function as
$$\delta[\mathcal{C}(\phi)] = \int \mathrm{D}\lambda \exp\!\left(\mathrm{i}\int \lambda\,\mathcal{C}(\phi)\right).$$
Now $\lambda$ is a new field, cooked up specifically to enforce the constraint $\mathcal{C}(\phi)$. So, here you have it, your theory now involves a path integral with a field with no kinetic terms
$$ Z = \int \mathrm{D}\phi\,\mathrm{D}\lambda \exp\!\left(-S[\phi]+\mathrm{i}\int\lambda\;\mathcal{C}(\phi)\right). $$

Another common instance is discrete gauge fields. Namely, if you have a gauge theory with a discrete gauge group, $G$, the gauge fields, $a$, have no curvature (see e.g. this phys.SE answer of mine) and in fact, they take values in the cohomology, $\mathrm{H}^1(M;G)$. So the path integral of a typical gauge theory with a discrete gauge field would be of the form
$$Z = \sum_{a\in\mathrm{H}^1(M;G)}\int\mathrm{D}\phi\ \exp\!\left(-S[\phi]+\int a\; w\right),$$
where $w$ is some topological invariant build out of the data that define the theory.

To make this more fun, combine 1. and 2. and come into contact with the paper you're struggling with, if the discrete group is $G=\mathbb{Z}_N$, you can write a $\mathbb{Z}_N$ gauge field (in the spirit of 2.) by considering a good old $\mathrm{U}(1)$ gauge field, $a$ and then forcing it to be a $\mathbb{Z}_N$ gauge field by enforcing the constraints
$$ \d{a} = 0 \qquad\text{&}\qquad \mathrm{exp}\!\left(\mathbb{i} N \oint a\right)=1.\tag{1}$$
Usually, this is achieved by coupling the $\mathrm{U}(1)$ gauge-field to a BF term
$$ \sim N \int b\ \d{a} \sim N \int \d{b}\ a,\tag{2}$$
as explained, e.g. in this phys.SE question and answer. However, you see that in [1], the same thing can be achieved by coupling $a$ to a conserved current $J$, (e.g. in eq. (2.1) of [1]) such that $\mathrm{d}\star J = 0$. Writing $K=\star J$ the coupling looks like
$$\sim N \int K\ a. \tag{3}$$
Locally you can represent $K$ as $\d{\kappa}$ and you see how (3) takes the form of (2), thus enforcing the constraints (1). You can morally view $K$ as a Lagrange multiplier, in the spirit of 1. enforcing the constraint, thus the path integral for this $\mathbb{Z}_N$ theory would look like
$$ Z = \int \mathrm{D}a\; \mathrm{D}K\ \exp\!\left(-S[a] + \mathrm{i}\; N \int K\ a\right),$$
where $S[a]$ is a typical action of a $\mathrm{U}(1)$ gauge field, e.g. as is the case in [1] Maxwell with a theta term.

Why is there no kinetic term?
In the case of [1], there is no kinetic term, simply because it is not allowed by symmetry/it would vanish. $J$ is a conserved current, or equivalently $K$ satisfies $\mathrm{d}K=0$, so any kinetic terms would be zero.
Physical interpretation
You should see the action (2.1) or equivalently (3.2) of [1] as representing an effective action of the gauged charge $N$ matter. To unpack what I mean by that, forget all these actions and let's try to build them from scratch. Take some fields with a $\mathrm{U}(1)$ symmetry, so they come with a conserved current $J$, or equivalently $K=\star J$ such that $\d K=0$. As before you should imagine $K$ as $\d{\kappa}$ in the absence of topological obstructions.
Now, you want to write an effective theory of this conserved quantity, so you start writing the most relevant terms. Had you had no massless quantities the most relevant thing you could write would be
$$ S_\text{eff} \sim \int K^2 = \int K\wedge \star K,$$
as it contains the least number of derivatives. But in the case of [1], you have coupled your theory to a $\mathrm{U}(1)$ gauge field. So there is something more relevant you can write. That is
$$ \sim \int K\ a.$$
That's exactly what they wrote, that is the $J^\mu a_\mu$ term in (2.1) or the $n_\mu a_\mu$ term in (3.2). Therefore, the full effective action at this scale is (keeping in mind the charge, $N$ of the matter fields)
$$S_\text{eff}[a,K] = S[a] + \mathrm{i}\; N\int K\ a.$$
Last step: why stop here? Since you have massless degrees of freedom, nothing stops you from integrating out the matter fields completely. That's what they are doing by integrating over $J$/summing over $n$. The final effective action, the response theory, is what remains after that
$$ S_{\text{eff}}[a] = -\log\int\mathrm{D}K \exp(-S_\text{eff}[a,K]).$$
The fact that the matter is condensing, is important to guarantee that $\kappa$ in $\int K\; a = \int \d\kappa\; a$ fluctuates wildly, thus allowing the Lagrange multiplier interpretation of the first section. This is explained in more detail in appendix A, section A.2 of [1].
The lattice version
The lattice version of the story is just trying to imitate the field-theoretic story I explained above. So if you discretise each of my equations you should end up with (3.1) and (3.2) of [1].

References:
[1] Theory of oblique topological insulators, arxiv:2206.07725


TITLE: Example of application of limit in daily life
QUESTION [2 upvotes]: We know that concept of limit plays a central role in calculus.I had find information in internet about the application of concept of limit in daily life but unfortunately I failed to find it. Can anyone give examples?
For instance, the derivative, integral and continuity require the concept of limit. I wonder is there any other application in daily life or certain field.

REPLY [0 votes]: For example the area of a circle can be calculated as a limit of areas of (regular, equilateral) polygons drawn inside the circle. The limit is that you send the number of vertices to infinity to approximate the circle as best as possible.


text:
\begin{document}
\maketitle

\begin{abstract} We introduce new Elmendorf constructions for equivariant categories and posets, and we prove that they are compatible with the classical topological one. Our constructions are more concrete than their model-categorical counterparts, and they give rise to new proofs of the Elmendorf theorems for equivariant categories and posets.
\end{abstract}

\tableofcontents

\section{Introduction}

From the standpoint of homotopy theory, there are many different models for topological spaces. Work of Kan \cite{Kan} and others (cf. \cite{GZ}, \cite{MaySimp}, and \cite{QuillenHA}) established that the homotopy theory of spaces is equivalent to the homotopy theory of simplicial sets. Later on, Thomason \cite{ThomasonCat} proved that the homotopy theory of simplicial sets is equivalent to the homotopy theory of small categories, and more recently, Raptis \cite{Raptis} proved that the homotopy theory of small categories is equivalent to the homotopy theory of posets. All told, there are Quillen equivalences
	\[
	\b{Top} \leftrightarrows \b{sSet} \rightleftarrows \b{Cat} \rightleftarrows \b{Pos}
	\]
between the categories of spaces, simplicial sets, small categories, and posets.

The same is true equivariantly. Let $G$ be a discrete group. Then by work of Bohmann-Mazur-Osorno-Ozornova-Ponto-Yarnall \cite{BMOOPY} and May-Stephan-Zakharevich \cite{MSZ} there are Quillen equivalences
	\[
	\b{Top}^G \leftrightarrows \b{sSet}^G \rightleftarrows \b{Cat}^G \rightleftarrows \b{Pos}^G
	\]
between the corresponding categories of left $G$-actions. Thus, $G$-spaces may be modeled by $G$-simplicial sets, small $G$-categories, and $G$-posets, but there are yet more models. Let $\s{O}_G$ denote the orbit category of $G$, i.e. the category of coset spaces $G/H$ and the $G$-equivariant maps between them. Let
	\[
	\b{Fun}(\s{O}_G^\op,\b{Top}) \leftarrow \b{Top}^G : \Phi
	\]
be the functor that sends a $G$-space $X$ to its presheaf of fixed point subspaces $X^H$. Then the functor $\Phi$ has an inverse
	\[
	C : \b{Fun}(\s{O}_G^\op , \b{Top}) \to \b{Top}^G
	\]
up to weak equivalence. Here, a weak equivalence of $G$-spaces is a $G$-map $f : X \to Y$ such that $f^H : X^H \to Y^H$ is a weak homotopy equivalence for all subgroups $H \subset G$, and weak equivalence of presheaves is a natural transformation $\lambda : \s{X} \Rightarrow \s{Y}$ such that each component of $\lambda$ is a weak homotopy equivalence. In this sense, $G$-spaces are also modeled by topological presheaves over $\s{O}_G$.

The original construction of $C$ is due to Elmendorf \cite{Elmendorf}, and thus the existence of homotopy inverse functors $C :  \b{Fun}(\s{O}_G^\op , \b{Top}) \rightleftarrows \b{Top}^G : \Phi$ is typically called Elmendorf's theorem. He defined $C$ using a two-sided bar construction, and Piacenza \cite{Piacenza} later recast things in model categorical terms, with $C$ being the derived left adjoint to $\Phi$. Elmendorf and Piacenza's constructions have distinct advantages. Piacenza's approach generalizes to simplicial, categorical, and posetal contexts, and it produces model-categorical proofs that $G$-simplicial sets, $G$-categories, and $G$-posets can be modeled by presheaves over $\s{O}_G$ (cf. \cite{BMOOPY}, \cite{MSZ}, and \cite{Stephan}). On the other hand, Elmendorf's construction has the virtue of being simple and explicit. It uses no heavy machinery, and it lends itself well to direct analysis (cf. \cite[\S4]{BHOspectra}).

The purpose of this paper is to give similarly direct Elmendorf constructions for $G$-categories and for $G$-posets. We shall define two functors (cf. \S\ref{subsec:catelm} and \S\ref{subsec:poselm})
	\[
	C : \b{Fun}(\s{O}_G^\op,\b{Cat}) \to \b{Cat}^G	\quad\t{and}\quad
	C : \b{Fun}(\s{O}_G^\op,\b{Pos}) \to \b{Pos}^G
	\]
and then prove that they have the expected properties. These functors are elementary. The $C$ functor for $G$-categories is a certain Grothendieck construction. The $C$ functor for $G$-posets is not simply the restriction of this functor, because it generally outputs $G$-preorders. Therefore we must rectify it. Surprisingly, the natural posetal quotient of a $G$-preorder does not usually have the correct equivariant homotopy type, so we use a version of Milnor's infinite join instead. This join construction gives a functorial posetal resolution of every preorder, and it may be of independent interest beyond this paper.

These constructions stand in sharp contrast to the model categorical analogues of $C$. The latter involve cofibrant resolution in the relevant presheaf categories, and while this is perfectly fine for theoretical considerations, these resolutions are somewhat inexplicit and sometimes difficult to compute. Our constructions have explicit formulas.

We now state our main results. Fix a discrete group $G$, not necessarily finite, and let $B = \abs{\cdot} \circ N : \b{Cat} \to \b{Top}$ denote the classifying space functor.

\begin{thmcomp}Suppose $\s{C} = \b{Cat}$ or $\b{Pos}$. Then for any orbit presheaf $\s{X} : \s{O}_G^\op \to \s{C}$, there is a zig-zag of weak $G$-equivalences between $CB\s{X}$ and $BC\s{X}$.
\end{thmcomp}

In the case $\s{C} = \b{Cat}$, this equivalence may be realized by a single map, and in the case $\s{C} = \b{Pos}$, it may be realized by a zig-zag of two maps. Thus, our Elmendorf constructions are direct lifts of the classical construction to the category of small categories and to the category of posets.

Next, let $\Phi$ denote the fixed point functor generically.

\begin{thmElm} Suppose $\s{C} = \b{Cat}$ or $\b{Pos}$. Then the functors
	\[
	C : \b{Fun}(\s{O}_G^\op,\s{C}) \rightleftarrows \s{C}^G : \Phi
	\]
are inverse up to weak equivalence.
\end{thmElm}

Here, a weak equivalence of categories or posets is a functor $F : \s{D} \to \s{E}$ such that $BF$ is a homotopy equivalence of spaces, a weak equivalence of presheaves is a natural transformation $\lambda : \s{X} \Rightarrow \s{Y}$ such that each component of $\lambda$ is a weak equivalence, and a weak equivalence of $G$-categories or $G$-posets is a $G$-functor $F : \s{D} \to \s{E}$ such that $BF$ is an equivalence of $G$-spaces. Thus, our Elmendorf constructions really are Elmendorf constructions in the usual sense, and they reestablish the equivalence between $\s{C}^G$ and $\b{Fun}(\s{O}_G^\op,\s{C})$ when $\s{C} = \b{Cat}$ or $\b{Pos}$.

We reiterate that the Elmendorf theorems for $G$-categories and $G$-posets were already proven using model-categorical arguments in \cite{BMOOPY} and \cite{MSZ}. The point of this paper is to give concrete formulas for the Elmendorf constructions, and to give new, elementary proofs of the Elmendorf theorems for $G$-categories and $G$-posets. We hope that our techniques will further facilitate the study of equivariant homotopy theory through the lens of $G$-categories and $G$-posets.

\subsection*{Organization} In \S\ref{sec:prelim}, we review some facts about $G$-categories, the Elmendorf theorem, and homotopy colimits. In \S\ref{sec:ElmCat} we introduce the categorical Elmendorf construction and then we prove it has the desired properties. In \S\ref{sec:ElmPos}, we explain how to resolve a $G$-preorder by a $G$-poset, and then we conclude with a discussion of the posetal Elmendorf construction.

\subsection*{Acknowledgements} It is a pleasure to thank Mike Hill, Peter May, and Ang\'{e}lica Osorno for many helpful conversations and for their incisive commentary. This work was partially supported by NSF Grant DMS--1803426.

\section{Preliminaries}\label{sec:prelim}

The goal of this paper is to lift the Elmendorf construction to $G$-categories and to $G$-posets. Our approach is motivated by two observations:
	\begin{enumerate}
		\item{}the Elmendorf construction is a homotopy colimit, and
		\item{}homotopy colimits are modeled by Grothendieck constructions.
	\end{enumerate}
Accordingly, we begin by establishing conventions on $G$-categories, and then we review Elmendorf's theorem and Thomason's homotopy colimit theorem.

\subsection{$G$-categories} Let $G$ be a discrete group, not necessarily finite. A (small) \emph{$G$-category} is a (small) category $\s{C}$, equipped with a homomorphism $\mu : G \to \b{Aut}(\s{C})$. Thus, for every $g \in G$, we have an invertible functor $g \cdot (-) = \mu(g) : \s{C} \to \s{C}$, and these action maps satisfy the usual associative and unital laws.

A \emph{$G$-functor} $F : \s{C} \to \s{D}$ between $G$-categories is a functor that preserves the action maps, i.e. $[g \cdot (-)] \circ F = F \circ [g \cdot (-)]$ for all $g \in G$. This means $gFx = Fgx$ and $gFf = Fgf$ for every object $x \in \s{C}$, morphism $f \in \s{C}$, and $g \in G$.

A \emph{$G$-natural transformation} between $G$-functors $F,H : \s{C} \rightrightarrows \s{D}$ is a natural transformation $\eta : F \Rightarrow H$ such that $[g \cdot (-)] \circ \eta = \eta \circ [g \cdot (-)]$ as whiskered natural transformations. This means $g\eta_x = \eta_{gx}$ for every $x \in \s{C}$ and $g \in G$.

The collection of all small $G$-categories, $G$-functors, and $G$-natural transformations forms a $2$-category, with componentwise vertical composition and the usual horizontal composition. We write
	\begin{enumerate}[label=(\roman*)]
		\item{}$\b{Cat}^G$ for the $1$-category of all small $G$-categories and $G$-functors,
		\item{}$\underline{\b{Cat}}^G$ for the $2$-category of all small $G$-categories, $G$-functors, and $G$-natural transformations, and
		\item{}$\b{Fun}^G(\s{C},\s{D})$ for the $1$-category of all $G$-functors and $G$-natural transformations from $\s{C}$ to $\s{D}$.
	\end{enumerate}

We shall frequently consider the fixed point subcategories of a $G$-category $\s{C}$. For any subgroup $H \subset G$, the \emph{$H$-fixed subcategory} $\s{C}^H$ is the subcategory of $\s{C}$ consisting of the $H$-fixed objects and the $H$-fixed morphisms between them. As one might expect, $H$-fixed points are representable. Let $(-)^\t{disc} : \b{Set} \to \b{Cat}$ be the discrete category functor, which sends a set $X$ to the category $X^\t{disc}$, whose object set is $X$, and which has no nonidentity morphisms. Then there is an induced functor $(-)^\t{disc} : \b{Set}^G \to \b{Cat}^G$, and an isomorphism
	\[
	\b{Fun}^G(G/H^\t{disc} , \s{C}) \stackrel{\cong}{\longrightarrow} \s{C}^H ,
	\]
induced by evaluating $G$-functors and $G$-natural transformations at $eH \in G/H^{\t{disc}}$.

The isomorphism above suggests a definition of the fixed point presheaf of $\s{C}$. Since $\underline{\b{Cat}}^G$ is a $2$-category, there is an enriched hom bifunctor
	\[
	\b{Fun}^G(-,-) : (\b{Cat}^G)^\op \times \b{Cat}^G \to \b{Cat},
	\]
which sends a pair of $G$-categories $(\s{C},\s{D})$	to the category $\b{Fun}^G(\s{C},\s{D})$ and a pair of $G$-functors $(F,H)$ to the composition functor $H \circ (-) \circ F$. We thus obtain a composite functor
	\[
	\b{Fun}^G(-,-) \circ [(-)^\t{disc} \times \t{id}] : \s{O}_G^\op \times \b{Cat}^G \to (\b{Cat}^G)^\op \times \b{Cat}^G \to \b{Cat},
	\]
and the transpose
	\[
	\Phi : \b{Cat}^G \to \b{Fun}(\s{O}_G^\op , \b{Cat})
	\]
is the fixed point functor.

For any $G$-category $\s{C}$, the presheaf $\Phi \s{C}$ sends $G/H \in \s{O}_G$ to $\b{Fun}^G(G/H^\t{disc} , \s{C})$ and $f : G/K \to G/H$ to the precomposition map $f^*$. Under the identification $\b{Fun}^G(G/H^\t{disc} , \s{C}) \cong \s{C}^H$, the morphism $r_a : G/K \to G/H$, which maps $eK$ to $aH$, is sent to the multiplication map $\s{C}^K \leftarrow \s{C}^H : a \cdot (-)$. For any $G$-functor $F : \s{C} \to \s{D}$, the corresponding map $\Phi F : \Phi \s{C} \Rightarrow \Phi \s{D}$ is postcomposition by $F$. Under the identification $\b{Fun}^G(G/H^\t{disc} , \s{C}) \cong \s{C}^H$, it corresponds to the restriction of $F$ to $H$-fixed objects and morphisms.

We think of a small $G$-category $\s{C}$ as a model for a $G$-space. To construct a $G$-space from $\s{C}$, we first apply the nerve functor $N : \b{Cat} \to \b{sSet}$ and then we apply the geometric realization functor $\abs{\cdot} : \b{sSet} \to \b{Top}$. We write $B = \abs{\cdot} \circ N$, and refer to $B\s{C}$ as the \emph{classifying space} of $\s{C}$. Homotopical notions for spaces are transported to categories along $B$. In particular, a $G$-functor $F : \s{C} \to \s{D}$ is a weak equivalence if $BF : B\s{C} \to B\s{D}$ is a weak $G$-equivalence, and a weak equivalence $\lambda : \s{X} \Rightarrow \s{Y}$ of presheaves is a natural transformation that is a levelwise weak equivalence. In light of the following proposition, a $G$-functor $F$ is a weak equivalence of $G$-categories if and only if $\Phi F$ is a weak equivalence of presheaves.

\begin{prop}\label{prop:fpsSet} Suppose $G$ is a (possibly infinite) discrete group, $H \subset G$ is a subgroup, and $K$ is a $G$-simplicial set. Then $\abs{K^H}$ is naturally isomorphic to $\abs{K}^H$.
\end{prop}

\begin{cor}\label{cor:fpCat} Suppose $G$ is a (possibly infinite) discrete group, $H \subset G$ is a subgroup, and $\s{C}$ is a $G$-category. Then $B(\s{C}^H)$ is naturally isomorphic to $(B\s{C})^H$.
\end{cor}

Proposition \ref{prop:fpsSet} and its corollary seem to be known, but we could not find proofs in the literature. When $G$ is a finite group, they hold because geometric realization preserves finite limits. It takes just a bit more work in the infinite case. The idea is that the space $\abs{K}^H$ is the subcomplex of $H$-fixed cells in $\abs{K}$, but we give a more categorical argument.

\begin{proof}[Proof of Proposition] The $H$-fixed points $K^H$ fit into an equalizer diagram
	\[
	\begin{tikzpicture}
		\node(KH) at (0,0) {$K^H$};
		\node(K) at (2,0) {$K$};
		\node(P) at (4.5,0) {$\prod_{h \in H} K$};
		\path[->]
		(KH) edge [above] node {$i$} (K)
		(K.15) edge [above] node {$\Delta$} (P.174)
		(K.-15) edge [below] node {$\mu$} (P.186)
		;
	\end{tikzpicture}
	\]
where $\Delta$ is the diagonal and the $h$-component of $\mu$ is the multiplication map $h \cdot (-) : K \to K$. Applying geometric realization yields a diagram
	\[
	\begin{tikzpicture}
		\node(KH) at (0,0) {$\abs{K^H}$};
		\node(K) at (2,0) {$\abs{K}$};
		\node(P) at (4.5,0) {$\abs{\prod_{h \in H}K}$};
		\node(P2) at (4.5,-2) {$\prod_{h \in H} \abs{K}$};
		\path[->]
		(KH) edge [above] node {$\abs{i}$} (K)
		(K.10) edge [above] node {$\abs{\Delta}$} (P.176)
		(K.-10) edge [below] node {$\abs{\mu}$} (P.184)
		(K.-40) edge [above right] node {$\Delta$} (P2.155)
		(K.-69) edge [below left] node {$\lambda$} (P2.163)
		(P) edge [right] node {$\kappa$} (P2)
		;
	\end{tikzpicture}
	\]
where the $h$-component of $\lambda$ is the multiplication map $\abs{h \cdot (-)} : \abs{K} \to \abs{K}$, and $\kappa$ is the canonical map. The top row of this diagram is an equalizer, because realization preserves finite limits. The morphism $\kappa$ is not a homeomorphism when $H$ is infinite, but $\kappa \circ \abs{\Delta} = \Delta$, and $\kappa \circ \abs{\mu} = \lambda$. Therefore $\Delta \circ \abs{i} = \lambda \circ \abs{i}$. It remains to show that $\abs{i}$ is an equalizer of $\Delta$ and $\lambda$.

So suppose $X$ is a space and $f : X \to \abs{K}$ is a continuous map such that $\Delta \circ f = \lambda \circ f$. Choose an element $x \in X$. By the uniqueness of the representation of elements of $\abs{K}$ in non-degenerate form (cf. \cite[\S14]{MaySimp}), we may write $f(x) = \abs{k_n,u_n}$, where $k_n \in K_n$ is non-degenerate, $u_n$ is an interior point of $\Delta^n$, and $\abs{k_n,u_n}$ is the class of $(k_n,u_n)$ in $\abs{K}$. Then $\abs{k_n,u_n} = \abs{hk_n,u_n}$ for every $h \in H$, and it follows that $k_n = hk_n$ for all $h \in H$, because $hk_n$ is also nondegenerate. Consequently,
	\[
	\abs{\Delta}(f(x)) =
	\abs{(k_n)_{h \in H},u_n} = \abs{(hk_n)_{h \in H},u_n}
	= \abs{\mu}(f(x)) ,
	\]
so that $\abs{\Delta} \circ f = \abs{\mu} \circ f$. By the universal property of $\abs{i}$, there is a unique morphism $\widetilde{f} : X \to \abs{K^H}$ such that $\abs{i} \circ \widetilde{f} = f$. This proves $\abs{i}$ is an equalizer of $\Delta$ and $\lambda$.
\end{proof}

We now turn to topology.

\subsection{The topological Elmendorf construction}\label{sec:topElm} Let $G$ be a compact Lie group, and let $J : \s{O}_G \to \b{Top}^G$ be the inclusion. Given any presheaf $\s{X} : \s{O}_G^\op \to \b{Top}$, the topological Elmendorf construction applied to $\s{X}$ is the two-sided bar construction
	\begin{align*}
		C\s{X}	&=	B(\s{X},\s{O}_G,J) .
	\end{align*}
This is the realization of the simplicial space, whose space of $q$-simplices is
	\[
	\coprod_{G/H_0,\dots,G/H_q \in \s{O}_G} \s{X}(G/H_q) \times \s{O}_G(G/H_{q-1},G/H_q) \times \cdots \times \s{O}_G(G/H_0,G/H_1) \times G/H_0 .
	\]
From a conceptual standpoint, the functor $C$ is the derived left adjoint to the fixed point functor $\Phi : \b{Top}^G \to \b{Fun}(\s{O}_G^\op,\b{Top})$, so one might hope that it is homotopy inverse to $\Phi$. This is precisely what Elmendorf proved.

\begin{thm}[\protect{\cite[Theorem 1 and its Corollary]{Elmendorf}}]\label{thm:Elmclass} The functors
	\[
	C : \b{Fun}(\s{O}_G^\op,\b{Top}) \rightleftarrows \b{Top}^G : \Phi
	\]
are inverse up to weak equivalence.
\end{thm}

We could try to lift Elmendorf's construction to $G$-categories and $G$-posets by mimicking the preceding bar construction, but this is ill-advised. The bar construction is a colimit, and colimits of categories are usually intractable. Instead, we shall use Thomason's homotopy colimit theorem. We will review Thomason's theorem in the next section, but for now, we note that Thomason's theorem does not quite apply to the Elmendorf construction $C\s{X}$ as written. Therefore we must repackage the data in $B(\s{X},\s{O}_G,J)$.

When $G$ is a discrete group, the spaces $\s{O}_G(G/H,G/K)$ and $G/H$ are discrete for all subgroups $K,H \subset G$. It follows that
	\[
	B_q(\s{X},\s{O}_G,J)
	\,\,\,\,
	\cong
	\!\!\!\!\!\!\!\!
	\coprod_{\tiny{\begin{array}{c}
	G/H_q \stackrel{f_q}{\leftarrow} \cdots \stackrel{f_1}{\leftarrow} G/H_0  \t{ in } \s{O}_G\\
	a_0 H_0 \in G/H_0
	\end{array}}}
	\!\!\!\!\!\!\!\!
	\s{X}(G/H_q) .
	\]
Now we move the elements $a_0 H_0 \in G/H_0$ into the indexing category.

\begin{defn}\label{defn:OG+} Let $\s{O}_{G,+}$ be the category whose objects are pairs $(G/H,aH)$, where $G/H \in \s{O}_G$ is an orbit and $aH \in G/H$ is a coset, and whose morphisms $f : (G/H,aH) \to (G/K , bK)$ are $G$-maps $f : G/H \to G/K$ such that $f(aH) = bK$. The group $G$ acts functorially on $\s{O}_{G,+}$ via
	\[
	g \cdot (G/H,aH) = (G/H,gaH)	\quad\t{and}\quad	g \cdot f = f ,
	\]
and there is a forgetful functor $p : \s{O}_{G,+} \to \s{O}_G$.
\end{defn}

We obtain a further homeomorphism
	\[
	\coprod_{\tiny{\begin{array}{c}
	G/H_q \stackrel{f_q}{\leftarrow} \cdots \stackrel{f_1}{\leftarrow} G/H_0 \t{ in } \s{O}_G, \\
	a_0 H_0 \in G/H_0
	\end{array}}}
	\!\!\!\!\!\!\!\!
	\s{X}(G/H_q) \,\,\,\,\,\,\,\,\,\,\,\, \cong \!\!\!\!\!\!\!\!\!\!\!\!\!\!
	\coprod_{\tiny{
	\begin{array}{c}
	(G/H_q,a_q H_q) \stackrel{f_q}{\leftarrow} \cdots \stackrel{f_1}{\leftarrow} (G/H_0,a_0 H_0) \\
	\t{ in } \s{O}_{G,+}
	\end{array}
	}}
	\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
	(\s{X} \circ p)(G/H_q, a_q H_q) ,
	\]
and it becomes a $G$-homeomorphism if we let $G$ act by permuting summands. The right hand side is $B_q(\s{X} \circ p , \s{O}_{G,+},*)$, and we arrive at the following conclusion.

\begin{prop}\label{prop:Elmhocolim} Suppose $G$ is a discrete group. For any $\s{X} : \s{O}_G^\op \to \b{Top}$, there is a $G$-homeomorphism between $C\s{X} = B(\s{X},\s{O}_G,J)$ and $B(\s{X} \circ p,\s{O}_{G,+},*)$.
\end{prop}

The space $B(\s{X} \circ p,\s{O}_{G,+},*)$ is the Bousfield-Kan model of $\t{hocolim}_{\s{O}_{G,+}^\op} (\s{X} \circ p)$  \cite[\S 12.5]{BousfieldKan}, which is precisely what Thomason's theorem concerns.

\subsection{Thomason's homotopy colimit theorem} We now review the Grothendieck construction and Thomason's theorem. Let  $\s{C}$ be a small category, and let $\b{Cat}$ denote the category of small categories. Recall that if $F : \s{C} \to \b{Cat}$ is a functor, then the \emph{Grothendieck construction} on $F$ is the small category $\int_{\s{C}} F$, whose objects are pairs $(c,x)$ such that $c \in \s{C}$ and $x \in Fc$, and whose morphisms $(c,x) \to (d,y)$ are pairs $(f,h)$ such that $f : c \to d$ in $\s{C}$ and $h : Ff(x) \to y$ in $Fd$. Composition is defined by $(f',h') \circ (f,h) = (f' \circ f, h' \circ Ff'(h))$, and $\t{id}_{(c,x)} = (\t{id}_c , \t{id}_x)$.

Thomason's homotopy colimit theorem states that Grothendieck constructions are categorical models for homotopy colimits.

\begin{thm}[\protect{\cite[Theorem 1.2]{Thomasonhocolim}}] Let $F : \s{C} \to \b{Cat}$ be a functor. Then there is a natural homotopy equivalence
	\[
	\eta : \t{hocolim}_{\s{C}} \, NF \stackrel{\simeq}{\longrightarrow} N \Big( \int_{\s{C}} F \Big) ,
	\]
where $N : \b{Cat} \to \b{sSet}$ is the nerve functor.
\end{thm}

We will need a precise description of Thomason's map going forward, and conveniently, Thomason also uses the Bousfield-Kan model of the homotopy colimit. Therefore we understand $\t{hocolim}_{\s{C}} \, NF$ to be the diagonal of the bisimplicial set, whose space of $q$-simplices is
	\[
	\coprod_{
	\tiny{
	\begin{array}{c}
	c_0 \leftarrow \cdots \leftarrow c_q
	\t{ in } \s{C}
	\end{array}
	}
	} NF(c_q) .
	\]
Under this model, the map $\eta : \t{hocolim}_{\s{C}} \, NF \to N(\int F)$ sends a $(q,q)$-simplex
	\[
	(\underbrace{c_0 \stackrel{f_1}{\leftarrow} \dots \stackrel{f_q}{\leftarrow} c_q}_{\t{in } \s{C}} , \underbrace{x_0 \stackrel{h_1}{\leftarrow} \cdots \stackrel{h_q}{\leftarrow} x_q}_{\t{in } F(c_q)})
	\]
to the $q$-simplex
	\[
	\Big( (c_0,F(f_1 \cdots f_q)(x_0) ) \stackrel{k_1}{\leftarrow} (c_1, F(f_2 \cdots f_q)(x_1)) \stackrel{k_2}{\leftarrow} \cdots \stackrel{k_q}{\leftarrow} (c_q , x_q) \Big) ,
	\]
where $k_i = (f_i , F(f_i \cdots f_q)(h_i))$.

We now have all the tools needed to prove the categorical Elmendorf theorem.

\section{The construction for $G$-categories}\label{sec:ElmCat}

With Proposition \ref{prop:Elmhocolim} and Thomason's theorem in tow, we now explain how to lift the Elmendorf construction to the category of small $G$-categories. We define the categorical Elmendorf construction, and then we establish its basic properties.

\subsection{The categorical Elmendorf construction}\label{subsec:catelm} Here is the definition.

\begin{defn}\label{defn:catElm} Let $G$ be a discrete group, and suppose $\s{X} : \s{O}_G^\op \to \b{Cat}$ is a presheaf of categories. The \emph{categorical Elmendorf construction} applied to $\s{X}$ is the Grothendieck construction
	\[
	C\s{X} = \int_{\s{O}_{G,+}^\op} (\s{X} \circ p) ,
	\]
where $p : \s{O}_{G,+} \to \s{O}_G$ is the forgetful functor (cf. Definition \ref{defn:OG+}).

We unwind this. For any $\s{X} : \s{O}_G^\op \to \b{Cat}$:
	\begin{enumerate}
		\item{}the objects of $C\s{X}$ are triples $(G/H,aH,x)$ such that $G/H \in \s{O}_G$, $aH \in G/H$, and $x \in \s{X}(G/H)$, and
		\item{}the morphisms of $\int (\s{X} \circ p)$ are pairs $(f,h) : (G/H,aH,x) \to (G/K,bK,y)$ such that $(G/H,aH) \leftarrow (G/K,bK) : f$ in $\s{O}_{G,+}$ and $h : f^* x \to y$ in $\s{X}(G/K)$, where $f^* = \s{X}(f)$.
	\end{enumerate}
Define a $G$-action on $\int (\s{X} \circ p)$ by setting $g \cdot (G/H,aH,x) = (G/H,gaH,x)$ and $g \cdot (f,h) = (f,h)$.

Now let $\lambda : \s{X} \Rightarrow \s{Y}$. Define a $G$-functor $C\lambda : C\s{X} \to C\s{Y}$ by setting
	\[
	C\lambda(G/H , aH, x) = (G/H, aH, \lambda_{G/H}(x)) \quad\t{and}\quad C\lambda(f,h) = (f,\lambda_{G/K}(h)),
	\]
where $(f,h) : (G/H, aH, x) \to (G/K, bK, y)$. This makes $C$ into a functor.
\end{defn}

By Thomason's theorem, we immediately obtain a \emph{nonequivariant} equivalence
	\[
	\abs{\eta} : C(B\s{X}) = \t{hocolim}_{\s{O}_{G,+}^\op}(B \circ \s{X} \circ p) \stackrel{\simeq}{\longrightarrow}  B \int_{\s{O}_{G,+}^\op} (\s{X} \circ p) = B(C\s{X}) ,
	\]
and we now prove that it respects the $G$-action.

\begin{thm}\label{thm:compElm} Let $G$ be a discrete group. For any orbit presheaf $\s{X} : \s{O}_G^\op \to \b{Cat}$, Thomason's map
	\[
	\abs{\eta} :  CB\s{X} \to BC\s{X}
	\]
is a weak $G$-equivalence.
\end{thm}

\begin{proof} We show how to identify the maps that $\eta$ induces on fixed points with other instances of $\eta$. Let $\s{X} : \s{O}_G^\op \to \b{Cat}$ be an orbit presheaf, let $K \subset G$ be a subgroup, and abbreviate $\s{O}_{G,+}^\op$ to $\s{O}$ and $(\s{O}_{G,+}^\op)^K$ to $\s{O}^K$. Then
	\[
	\Big[ \int_{\s{O}} (\s{X} \circ p) \Big]^K = \int_{\s{O}^K} (\s{X} \circ p)\Big|_{\s{O}^K} ,
	\]
because these categories are both the full subcategory of $\int (\s{X} \circ p)$ spanned by the triples $(G/H, aH , x)$ such that $G/H \in \s{O}_G$, $aH \in (G/H)^K$, and $x \in \s{X}(G/H)$. Consequently,
	\[
	\Big[ N \int_{\s{O}} (\s{X} \circ p) \Big]^K = N \Big[ \int_{\s{O}^K} (\s{X} \circ p)\Big|_{\s{O}^K} \Big] .
	\]

Next, we equip $\t{hocolim}_{\s{O}} (N \circ \s{X} \circ p) = \t{diag} \, B_*(N \circ \s{X} \circ p , \s{O}_{G,+} , *) $ with a $G$-action as follows. Let $G$ act on the space
	\begin{align*}
	B_q(N \circ \s{X} \circ p , \s{O}_{G,+} , *)
	&=
	\!\!\!\!\!\!\!\!
	\coprod_{\tiny{\begin{array}{c}
		(G/H_q , a_q H_q) \leftarrow \cdots \leftarrow (G/H_0,a_0 H_0)	\\
		\t{in } \s{O}_{G,+}
		\end{array}}}
		\!\!\!\!\!\!\!\!
		(N \circ \s{X})(G/H_q)
	\end{align*}
of $q$-simplices by permuting summands. This simplicial action induces a $G$-action on $\t{diag} \, B_*(N \circ \s{X} \circ p , \s{O}_{G,+} , *)$, whose geometric realization is the $G$-action on the topological Elmendorf construction $CB\s{X}$. Moreover,
	\[
	[\t{diag} \, B_*(N \circ \s{X} \circ p , \s{O}_{G,+} , *) ]^K
	=
	\t{diag} \, B_* ( (N \circ \s{X} \circ p)|_{\s{O}^K} , \s{O}^K , *)
	\]
because on either side, the $q$-simplices are tuples
	\[
	\Big(
	\underbrace{(G/H_0 , a_0 H_0) \to \cdots \to (G/H_q, a_q H_q)}_{\t{in } \s{O}_{G,+}}
	\, , \,
	\underbrace{x_0 \leftarrow \cdots \leftarrow x_q}_{\t{in } \s{X}(G/H_q)}
	\Big)
	\]
such that $a_0 H_0 ,\dots , a_q H_q$ are all $K$-fixed.

Thomason's map $\eta : \t{hocolim} (N \circ \s{X} \circ p) \to N \int (\s{X} \circ p)$ is $G$-equivariant,  and inspecting his formula shows that there is a commutative square
	\[
	\begin{tikzpicture}
		\node(A) at (6,0) {$\Big[ N \int_{\s{O}} (\s{X} \circ p) \Big]^K$};
		\node(B) at (0,0) {$[\t{hocolim}_{\s{O}} (N \circ \s{X} \circ p)]^K$};
		\node(C) at (6,-1.5) {$N\Big[ \int_{\s{O}^K} (\s{X} \circ p)\Big|_{\s{O}^K} \Big]$};
		\node(D) at (0,-1.5) {$\t{hocolim}_{\s{O}^K} (N \circ \s{X} \circ p)|_{\s{O}^K}$};
		\path[->]
		(A) edge [right] node {$=$} (C)
		(B) edge [left] node {$=$} (D)
		(B) edge [above] node {$\eta^K$} (A)
		(D) edge [below] node {$\eta$} (C)
		;
	\end{tikzpicture} .
	\]
The bottom map is an equivalence by Thomason's homotopy colimit theorem, which implies that $\eta^K$ is an equivalence for all subgroups $K \subset G$. Applying the geometric realization functor $\abs{\cdot} : \b{sSet} \to \b{Top}$ and using Proposition \ref{prop:fpsSet} to commute $(-)^K$ with $\abs{\cdot}$ shows that
	\[
	\abs{\eta}^K : (CB\s{X})^K \to (BC\s{X})^K
	\]
is a homotopy equivalence for all $K$. Hence $\abs{\eta}$ is a weak $G$-equivalence.
\end{proof}

\subsection{Elmendorf's theorem for $G$-categories}

One would hope that the categorical Elemendorf construction $C : \b{Fun}(\s{O}_G^\op,\b{Cat}) \to \b{Cat}^G$ is homotopy inverse to taking fixed points, but this is not automatic from Theorem \ref{thm:compElm}. Therefore we shall prove it.

\begin{thm}\label{thm:catcoal} For any discrete group $G$, the functors
	\[
	C : \b{Fun}(\s{O}_G^\op,\b{Cat}) \rightleftarrows \b{Cat}^G : \Phi
	\]
are inverse up to natural weak equivalence.
\end{thm}

The required transformations $C \circ \Phi \Rightarrow \t{id}$ and $\Phi \circ C \Rightarrow \t{id}$ are constructed in Propositions \ref{prop:wePhiC} and \ref{prop:weCPhi}.

\begin{prop}\label{prop:wePhiC} There is a natural weak equivalence $\ve : \Phi \circ C \Rightarrow \t{id}$.
\end{prop}

\begin{proof} Let $\s{X} : \s{O}_G^\op \to \b{Cat}$ be an orbit presheaf, and $L \subset G$ be a subgroup. The $L$-fixed subcategory of $(C\s{X})^L$ is the full subcategory of $C\s{X}$ spanned by the triples $(G/H,aH,x)$ such that $aH \in (G/H)^L$. Construct a functor
	\[
	(\ve_\s{X})_L : (C\s{X})^L \to \s{X}(G/L)
	\]
as follows:
	\begin{enumerate}
		\item{}Given any $L$-fixed object $(G/H,aH,x)$, let $r_a : G/L \to G/H$ be the $G$-map sending $eL$ to $aH$, and let $\s{X}(G/L) \leftarrow \s{X}(G/H) : r_a^*$ be its image under $\s{X}$. We define
			\[
			(\ve_{\s{X}})_L(G/H,aH,x) \stackrel{\t{def}}{=} r_a^* x.
			\]
		\item{}Given any morphism $(f,h) : (G/H,aH,x) \to (G/K,bK,y)$ between $L$-fixed triples, let $r_a : G/L \to G/H$ and $r_b : G/L \to G/K$ be as above, and note that $f \circ r_b = r_a$. We define
			\[
			(\ve_{\s{X}})_L(f,h) \stackrel{\t{def}}{=} r_b^* h : r_a^* x = r_b^* f^* x \to r_b^* y.
			\]
	\end{enumerate}
Straightforward checks show that $(\ve_{\s{X}})_L$ is natural in $L$ and $\s{X}$, i.e. $\ve$ is a natural transformation $\ve : \Phi \circ C \Rightarrow \t{id}_{\b{Fun}(\s{O}_G^\op,\b{Cat})}$.

It remains to check that $\ve_{\s{X}} : \Phi C \s{X} \Rightarrow \s{X}$ is a weak equivalence for every $\s{X}$. We shall construct a homotopy inverse to $\ve_L = (\ve_{\s{X}})_L : (C\s{X})^L \to \s{X}(G/L)$ for every $G/L$. Given any object $x \in \s{X}(G/L)$, let $\eta_L(x) = (G/L,eL,x) \in (C\s{X})^L$, and given any morphism $f : x \to y$ in $\s{X}(G/L)$, let $\eta_L(f) = (\t{id}_{G/L} , f)$. Then $\eta_L$ is a functor, and $\ve_L \circ \eta_L = \t{id}$. The composite $\eta_L \circ \ve_L$ is not the identity, but
	\[
	( r_a, \t{id}_{r_a^* x} ) : (G/H,aH,x) \to (G/L , eL , r_a^* x) = \eta_L \ve_L(G/H,aH,x)
	\]
defines a natural transformation $\t{id} \Rightarrow \eta_L \circ \ve_L$. Therefore the classifying space functor $B : \b{Cat} \to \b{Top}$ sends $\ve_L$ and $\eta_L$ to homotopy inverse functions. It follows that $B\ve_L$ is a homotopy equivalence for all $L$, and that $\ve : \Phi C \s{X} \Rightarrow \s{X}$ is a levelwise weak equivalence in $\b{Fun}(\s{O}_G^\op , \b{Cat})$.
\end{proof}

\begin{rem} As with the topological Elemendorf construction, the map $\eta_L$ does not vary naturally in $L$. Nevertheless, $\ve$ is a natural weak equivalence.
\end{rem}

\begin{prop}\label{prop:weCPhi} There is a natural weak equivalence $\t{ev} : C \circ \Phi \Rightarrow \t{id}$.
\end{prop}

\begin{proof} Suppose $\s{C}$ is a $G$-category, and consider $C \Phi \s{C}$. If $(G/H,aH,x)$ is an object of $C \Phi \s{C}$, then $x \in \s{C}^H$, and if $(f,h) : (G/H,aH,x) \to (G/K,bK,y)$ is a morphism of $C \Phi \s{C}$, then $h \in \s{C}^K$. Define a $G$-functor $\t{ev}_{\s{C}} : C \Phi \s{C} \to \s{C}$ by
	\[
	\t{ev}_{\s{C}}(G/H,aH,x) = ax	\quad\t{and}\quad	\t{ev}_{\s{C}}\Big( (f,h) : (G/H,aH,x) \to (G/K,bK,y) \Big) = bh.
	\]
One readily checks that this defines a natural transformation $\t{ev} : C \circ \Phi \Rightarrow \t{id}_{\b{Cat}^G}$, and that $\Phi \t{ev}_{\s{C}} = \ve_{\Phi \s{C}}$. The natural transformation $\ve_{\Phi \s{C}}$ is a weak equivalence by Proposition \ref{prop:wePhiC}, and it follows that $\t{ev}_{\s{C}}$ is also a weak equivalence, because $\Phi$ creates such morphisms. Therefore $\t{ev}$ is a natural weak equivalence.
\end{proof}

\subsection{Example: equivariant universal spaces}\label{app:univ} With the basic properties of the categorical Elmendorf construction established, we now consider an example. Equivariant universal spaces arise from the simplest orbit presheaves, and testing on them reveals a fundamental issue with the posetal Elmendorf construction.

 Let $\s{F}$ be a family of subgroups of $G$, and let $\s{X}_{\s{F}} : \s{O}_G^\op \to \b{Cat}$ be defined by
	\[
	\s{X}_{\s{F}}(G/H) = \left\{
	\begin{array}{cl}
		*	&\t{if } H \in \s{F}	\\
		\varnothing	&\t{if } H \notin \s{F}
	\end{array}
	\right. ,
	\]
where $*$ is the terminal category and $\varnothing$ is the initial one. The closure of $\s{F}$ under subconjugacy ensures that $\s{X}_{\s{F}}$ is a well-defined presheaf of categories.

Now regard a preorder $P$ as a category with at most one morphism $x \to y$ for any $x,y \in P$. Given such a category $\s{P}$, we recover a preorder on $\t{Ob}(\s{P})$ by declaring $x \leq y$ if there is a morphism $x \to y$. Then $\s{X}_{\s{F}}$ is actually a presheaf of posets, but this does not quite imply that $C\s{X}$ is a poset. Indeed, applying the Elmendorf construction to $\s{X}_{\s{F}}$ yields a \emph{$G$-preorder} $C\s{X}_{\s{F}}$, whose underlying $G$-set is
	\[
	C\s{X}_{\s{F}} = \coprod_{H \in \s{F}} G/H ,
	\]
and whose order relation is
	\[
	(H,aH) \leq (K,bK)	\quad\t{if and only if}\quad	aHa^{-1} \supset bKb^{-1} .
	\]
Specializing to the case $\s{F} = \{e\}$, we obtain
	\[
	C\s{X}_{\{e\}} = G/e
	\]
with $x \leq y$ for all $x,y \in G/e$. This is the standard categorical model of $EG$.

On the other hand, the categorical Elmendorf construction does send presheaves of preorders to $G$-preorders.

\begin{lem}\label{lem:Cpreord} If $\s{X} : \s{O}_G^\op \to \b{Cat}$ is valued in preorders, then $C\s{X}$ is a $G$-preorder.
\end{lem}

The proof is straightforward. This lemma is the starting point for our posetal Elmendorf construction.

\section{The construction for $G$-posets}\label{sec:ElmPos}

In this section, we explain how to lift Elmendorf's construction to the category of $G$-posets. The natural guess is to restrict the categorical Elmendorf construction $C\s{X} \cong \int (\s{X} \circ p)$ to presheaves of posets. The result is guaranteed to be a $G$-preorder by Lemma \ref{lem:Cpreord}, but it need not be a $G$-poset, as demonstrated by the presheaves $\s{X}_{\s{F}}$ in \S\ref{app:univ}. Therefore we must devise a method for converting $G$-preorders into $G$-posets, without changing their equivariant homotopy type.

\subsection{Posetal quotients}\label{sec:posquot} There is an adjunction $\ol{(-)} : \b{Preord} \rightleftarrows \b{Pos} : i$ between the category of preorders and the category of posets. The right adjoint $i$ is the inclusion functor, and the left adjoint $\ol{(-)}$ sends a preorder to its posetal quotient. This is the poset obtained by taking the quotient of a preorder $(P,\leq)$ by the equivalence relation
	\[
	x \sim y	\quad\t{if and only if}\quad		x \leq y \t{ and } y \leq x ,
	\]
and then declaring $[x] \leq [y]$ if $x \leq y$ for some choice of representatives.

For any preorder $P$, there is a quotient map $\pi : P \to \ol{P}$, and choosing a set of representatives for $\sim$ defines a section $P \leftarrow \ol{P} : s$. Thus $\pi \circ s = \t{id}$. The composite $s \circ \pi$ is not the identity, but for any $x \in P$, we have $x \leq s[x] \leq x$, i.e. there are natural transformations $\t{id}_P \Rightarrow s \circ \pi \Rightarrow \t{id}_P$. On passage to classifying spaces, we obtain a homotopy equivalence $BP \simeq B\ol{P}$.

Unfortunately, the equivalence $BP \simeq B\ol{P}$ does not hold equivariantly.

\begin{cex} Consider the $G$-preorder $P = C \s{X}_{\{e\}}$ from \S\ref{app:univ}. Then $\ol{P} = *$, and applying the classifying space functor gives $BP \simeq EG$ and $B\ol{P} = *$.
\end{cex}

We need another construction. In keeping with the rest of homotopy theory, we avoid taking a quotient and instead construct a resolution.

\subsection{Posetal resolutions}\label{subsec:Milnor} We now explain how to resolve a preorder by a poset, in such a way that preserves equivariant homotopy types.

By way of motivation, consider the space $EG$ once more. Then
	\[
	EG \simeq B(*,G,G) = B C \s{X}_{\{e\}},
	\]
but the space $EG$ can also be modeled by the Milnor join
	\[
	EG \simeq \t{colim}\, \underbrace{G * G * \cdots * G}_{n \t{ times}} .
	\]
When $G = \{g_1,\dots,g_n \}$ is a finite group, the Milnor join is the classifying space of the countably infinite poset
	\[
	\begin{tikzpicture}
		\node(10) at (1,0) {$g_1$};
		\node(20) at (2,0) {$g_2$};
		\node(30) at (3,0) {$\dots$};
		\node(40) at (4,0) {$g_n$};
		\node(11) at (1,1) {$g_1$};
		\node(21) at (2,1) {$g_2$};
		\node(31) at (3,1) {$\dots$};
		\node(41) at (4,1) {$g_n$};
		\node(12) at (1,2) {$g_1$};
		\node(22) at (2,2) {$g_2$};
		\node(32) at (3,2) {$\dots$};
		\node(42) at (4,2) {$g_n$};
		\node(13) at (1,3) {};
		\node(23) at (2,3) {};
		\node(33) at (3,3) {};
		\node(43) at (4,3) {};
		\node(14) at (1,3.5) {$\vdots$};
		\node(24) at (2,3.5) {$\vdots$};
		\node(34) at (3,3.5) {$\vdots$};
		\node(44) at (4,3.5) {$\vdots$};
		\path[-]
		(10)	edge node {} (11)
			edge node {} (21)
			edge node {} (41)
		(20)	edge node {} (11)
			edge node {} (21)
			edge node {} (41)
		(40)	edge node {} (11)
			edge node {} (21)
			edge node {} (41)

		(11)	edge node {} (12)
			edge node {} (22)
			edge node {} (42)
		(21)	edge node {} (12)
			edge node {} (22)
			edge node {} (42)
		(41)	edge node {} (12)
			edge node {} (22)
			edge node {} (42)

		(12)	edge node {} (13)
			edge node {} (23)
			edge node {} (43)
		(22)	edge node {} (13)
			edge node {} (23)
			edge node {} (43)
		(42)	edge node {} (13)
			edge node {} (23)
			edge node {} (43)
		;
	\end{tikzpicture} ,
	\]
where each vertex $g_i$ represents an element of the poset, and each upward edge represents an order relation $\leq$. The classifying space of the first row is $G$, the classifying space of the first two rows is the join $G*G$, and so on.

Formally, the poset above is the set $MG = G \times \bb{N}$ equipped with the relation
	\[
	(x,m) < (y,n)	\quad\t{if and only if}\quad		m<n .
	\]
The classifying space $BMG$ is nonequivariantly contractible because $MG$ is a filtered poset (cf. \cite[\S1]{QuillenHA}), and the group $G$ acts freely on $BMG$ because it acts freely on $MG$. Therefore $BMG \simeq EG$. The first coordinate projection $\pi : MG \to C\s{X}_{\{e\}}$ is order-preserving, and it induces an equivalence on classifying spaces. Therefore we regard $MG$ as a posetal resolution of $C\s{X}_{\{e\}}$.

We now generalize this $M$ construction to arbitrary $G$-preorders.

\begin{defn}\label{defn:Milnor} Suppose $(P,\leq)$ is a $G$-preorder. We define $MP$ to be the set
	\[
	MP = P \times \bb{N}
	\]
equipped with the partial order
	\begin{center}
		$(x,m) < (y,n)$ if and only if $x \leq_P y$ and $m < n$.
	\end{center}
We define a $G$-action on $MP$ by $g \cdot (x,m) = (gx,m)$ for all $g \in G$ and $(x,m) \in MP$. This makes $MP$ into a $G$-poset.

Next, given any map $f : P \to Q$ of $G$-preorders, we define
	\[
	Mf = f \times \t{id} : MP \to MQ.
	\]
This makes $M$ into a functor $M : \b{Preord}^G \to \b{Pos}^G$.
\end{defn}

The first coordinate projection $\pi : MP \to P$ is a natural order-preserving $G$-map, and we now prove that is a resolution of $P$. We begin with the nonequivariant result. In the following argument, we understand a \emph{subposet} of a poset $Q$ to be a subset $S \subset Q$ equipped with the restriction of $Q$'s order relation, i.e. if $x,y \in S$, then $x \leq_S y$ if and only if $x \leq_{Q} y$.

\begin{prop}\label{prop:Mpieq} For any nonequivariant preorder $P$, the projection $\pi : MP \to P$ induces a weak equivalence $B\pi : BMP \to BP$.
\end{prop}

\begin{proof} We compute the comma categories of $\pi : MP \to P$ and then apply Quillen's Theorem A. Consider the comma category $x \downarrow \pi$ for some $x \in P$. It can be identified with the subposet $\{ (y,n) \in MP \, | \, x \leq y \t{ in } P\} \subset MP$, and the latter contains $\pi^{-1}[x] = \{(y,n) \in MP \, | \, x \leq y \leq x \t{ in } P\}$ as a subposet. Here $[x]$ denotes the equivalence class of $x$ under the relation in \S\ref{sec:posquot}. We claim that the inclusion $i : \pi^{-1}[x] \hookrightarrow x \downarrow \pi$ is a homotopy equivalence.

Define a monotone map $r : x \downarrow \pi \to \pi^{-1}[x]$ by
	\[
	r(y,n) = \left\{
		\begin{array}{cl}
			(y,n)	&	\t{if } y \in [x] \\
			(x,n)	&	\t{if } y \notin [x]
		\end{array}
		\right. .
	\]
Then $r \circ i = \t{id} : \pi^{-1}[x] \to \pi^{-1}[x]$ and $Br \circ Bi$ is the identity on $B(\pi^{-1}[x])$. The composite $i \circ r$ is not the identity, but it is connected to the identity through a zig-zag of inequalities. Let $s : x \downarrow \pi \to x \downarrow \pi$ be the monotone map defined by
	\[
	s(y,n) = \left\{
		\begin{array}{cl}
			(y,n)	&	\t{if } y \in [x] \\
			(y,n+1)	&	\t{if } y \notin [x]
		\end{array}
		\right. .
	\]
Then $(y,n) \leq s(y,n) \geq ir(y,n)$ for all $(y,n) \in x \downarrow \pi$. Therefore $Bi \circ Br$ is homotopic to the identity on $B(x \downarrow \pi)$, and therefore $Bi : B(\pi^{-1}[x]) \to B(x \downarrow \pi)$ is a homotopy equivalence.

Now for any $(y,m) , (z,n) \in \pi^{-1}[x]$, we have the inequalities $y \leq x \leq z$, and therefore $(y,m) < (z,n)$ if and only if $m < n$. It follows $\pi^{-1}[x]$ is filtered, because any two elements $(y,m)$ and $(z,n)$ are bounded above by $(x,\max(m,n)+1)$. Therefore $B(x \downarrow \pi) \simeq B(\pi^{-1}[x]) \simeq *$, and Quillen's Theorem A implies that $B\pi : BMP \to BP$ is a homotopy equivalence.
\end{proof}

Now we boost Proposition \ref{prop:Mpieq} up to an equivariant result. Unlike the posetal quotient $\pi : P \to \ol{P}$, the projection map $\pi : MP \to P$ plays well with the $G$-action.

\begin{thm}\label{thm:MGeq} Let $G$ be a discrete group. For any $G$-preorder $P$, the natural projection $\pi : MP \to P$ is a weak equivalence of $G$-preorders.
\end{thm}

\begin{proof} For any subgroup $H \subset G$, there is an equality $(MP)^H = M(P^H)$ of posets, and a commutative square
	\[
	\begin{tikzpicture}
		\node(A) at (0,0) {$(MP)^H$};
		\node(B) at (3,0) {$P^H$};
		\node(C) at (0,-1.5) {$M(P^H)$};
		\node(D) at (3,-1.5) {$P^H$};
		\path[->]
		(A) edge [above] node {$\pi^H$} (B)
		(C) edge [below] node {$\pi$} (D)
		(A) edge [left] node {$=$} (C)
		(B) edge [right] node {$=$} (D)
		;
	\end{tikzpicture} .
	\]
By Proposition \ref{prop:Mpieq}, the bottom map is an equivalence on classifying spaces, and therefore the top map is, too. Thus $\pi^H$ is an equivalence for all subgroups $H \subset G$, i.e. $\Phi \pi$ is an equivalence, and therefore $\pi$ is also an equivalence because $\Phi$ creates these morphisms.\end{proof}

\begin{cor}The functors $M : \b{Preord}^G \rightleftarrows \b{Pos}^G : \t{inc}$ are inverse up to natural weak equivalence.
\end{cor}

\subsection{The Elmendorf theorem for $G$-posets}\label{subsec:poselm}

We conclude by discussing Elmendorf's theorem for $G$-posets. For clarity, we shall write $C_{cat}$ for the categorical Elmendorf construction and $C_{pos}$ for the posetal one.

\begin{defn} Let $G$ be a discrete group, and suppose $\s{X} : \s{O}_G^\op \to \b{Pos}$ is a presheaf of posets. The \emph{posetal Elmendorf construction} applied to $\s{X}$ is
	\[
	C_{pos}\s{X} = M C_{cat} \s{X},
	\]
where $C_{cat}$ and $M$ are the functors from Definitions \ref{defn:catElm} and \ref{defn:Milnor}.
\end{defn}

Unwinding everything shows that the underlying $G$-set of $C_{pos}\s{X}$ is
	\[
	\coprod_{G/H \in \s{O}_G} G/H \times \s{X}(G/H) \times \bb{N},
	\]
with $G$ acting on the $G/H$ factors only. Its elements are quadruples $(G/H,aH,x,m)$, where $G/H \in \s{O}_G$, $aH \in G/H$, $x \in \s{X}(G/H)$, and $m \in \bb{N}$. The partial order on $C_{pos}\s{X}$ is defined by
	\[
	(G/H,aH,x,m) < (G/K,bK,y,n)	\quad\t{if and only if}\quad
	\left\{
		\begin{array}{l}
			aHa^{-1} \supset bKb^{-1},	\\
			r_{b^{-1}a}^*x \leq y \t{ in } \s{X}(G/K),	\\
			\t{and } m < n
		\end{array}
	\right.
	\]
where $r_{b^{-1}a} : G/K \to G/H$ sends $eK$ to $b^{-1}aH$.

As with the categorical Elmendorf construction, the posetal Elmendorf construction is compatible with the original.

\begin{thm}\label{thm:Elmpostop} Let $G$ be a discrete group. For any orbit presheaf $\s{X} : \s{O}_G^\op \to \b{Pos}$, there is a zig-zag of natural weak $G$-equivalences between $CB\s{X}$ and $BC_{pos} \s{X}$.
\end{thm}

\begin{proof}The maps
	\[
	CB\s{X} \stackrel{\abs{\eta}}{\longrightarrow} B \Big( \int_{\s{O}_{G,+}} (\s{X} \circ p) \Big) = BC_{cat}\s{X} \stackrel{B\pi}{\longleftarrow} BC_{pos}\s{X}
	\]
are both weak $G$-equivalences by Theorems \ref{thm:compElm} and \ref{thm:MGeq}.
\end{proof}

Moreover, the posetal Elmendorf theorem holds for $C_{pos}$.

\begin{thm}\label{thm:Elmpos} For any discrete group $G$, the functors
	\[
	C_{pos} : \b{Fun}(\s{O}_G^\op , \b{Pos} ) \rightleftarrows \b{Pos}^G : \Phi
	\]
are inverse up to natural weak equivalence.
\end{thm}

\begin{proof}There are equivalences
	\[
	\Phi C_{pos} \s{X} = \Phi MC_{cat} \s{X} \stackrel{\Phi \pi}{\longrightarrow} \Phi C_{cat} \s{X} \stackrel{\ve}{\longrightarrow} \s{X}
	\]
by Theorem \ref{thm:MGeq} and Proposition \ref{prop:wePhiC}, and there are equivalences
	\[
	C_{pos}\Phi\s{X} = M C_{cat} \Phi\s{X} \stackrel{\pi}{\longrightarrow} C_{cat} \Phi\s{X} \stackrel{\t{ev}}{\longrightarrow} \s{X}
	\]
by Theorem \ref{thm:MGeq} and Proposition \ref{prop:weCPhi}.
\end{proof}


TITLE: Do photons emitted from different color filters leave varying size marks on a screen when fired?
QUESTION [3 upvotes]: I have read that photon "size" is proportion to wavelength, but I've also read that photon "size" isn't exactly a thing. So does this mean that different colors, which have different wavelengths and thus different "size" photons, will leave different size marks on a light sensitive screen when fired?
By "firing" photons, I mean the technology that results in a build up pattern of circular points, like here (although this one is with electrons):
https://www.youtube.com/watch?v=hv12oB_uyFs
And by different color filters, I mean placing different colored filters in front of a light source, like in this video:
https://www.youtube.com/watch?v=kcSYV8bJox8
So when you swap filter colors in front of whatever light source is being used to emit photons in a build up pattern experiment, do the circular dots that build up, have different sizes (or densities/intensities) dependent on the color of the filter used?
Or would it just be the build up pattern itself (like the interference pattern in a double slit experiment) that is different, and the individual build up marks themselves are all identical regardless of any colored filter that might be in front of the light?

REPLY [1 votes]: Well, first of all, filters do not create photons of a particular wavelength (there are substances that can absorb light at one frequency and emit another frequency, but that's another phenomenon). A filter simply is selective about what light it lets through, and will let through light of a particular wavelength if that wavelength was already present in the incident beam.
Second, the wavelengths of visible light are microscopic. You're not going to be able to see with your naked eye anything on that scale. Generally, patterns from shining a light beam are from vast numbers of photons. If a single photon is creating a visible dot, it's because there is a cascading reaction where the photon interacting with one atom sets off a chain reaction that affects a macroscopic area, not because the photon interacted directly with that entire area.
Third, a photon interacts with only one atom. Its wavelength will affect how many atoms it can interact with; a larger wavelength means more atoms that it could have interacted with. But it doesn't affect how many it does interact with. You can kind of think of it as a skier who's weaving back and forth as they go down a slope. A larger wavelength is like the skier weaving back and forth a larger distance. There's a larger area over which they can crash into a tree, but once they crash into a tree, they're going to stop and not crash into any more.
A larger wavelength will generally mean more dispersion, so if you try to focus a beam of photons to a single point, a larger wavelength will result in a larger spread, but less average intensity per unit area over the area it spreads out over. If you're just taking a light bulb and passing it through a filter, this doesn't apply, as light bulbs don't aim photons in a narrow beam.


TITLE: Why aren't proceedings from ICM 2014 on mathscinet?
QUESTION [25 upvotes]: Articles from the Proceedings of the International Congress of Mathematicians, Seoul, 2014 don't appear to be on Mathscinet. Why is this?
(Someone pointed this out to me recently, and I was reminded of it today when I tried to cite a lecture.)

REPLY [3 votes]: This is just a comment, in community-wiki format to make it easier to compactify links.
To expand on Ed Dunne's reply, it may be helpful to note that each ICM organizing committee makes its own arrangements about formal publication of proceedings.   For the Seoul ICM see Seoul and for PDF files of their four volumes of proceedings see here.   Presumably in due time all of this will be incorporated into the ICM site, where one can find older proceedings:  ICM.
Nowadays most people post earlier manuscripts on the arXiv, but of course they are more troublesome to track down.   In any case, the ICM databases of speakers and sessions are useful because published proceedings are usually organized into several volumes.