TITLE: Hall and Knight question
QUESTION [3 upvotes]: If n is any positive integer show that the integral part of $$(3+\sqrt7)^n$$is a odd number
I have no idea how to begin this problem but it is given in the chapter of binomial theorem so I hope that it is found using that only

REPLY [2 votes]: 'I'to denote the integral and 'f'to denote the fractional part of $(3+√7)^n$
Now $(3-√7)^n$ is less than 1 and a proper fraction let's
denote it by f'
$(3+√7)^n=3^n+ C_13^{n-1}√7......$
$3-√7)^n=3^n-C_13^{n-1}√7........$
As you can see when we add them the irrational terms cancel out.
$(3+√7)^n+(3-√7)^n$=I+f+f'= even integer
But since f and f' are proper fractions there are some must be 1
Hence we conclude that it's integral part is odd.


TITLE: How to represent the reduced mass in the curved space form, while studying a two-body system in GR?
QUESTION [1 upvotes]: The motion of a two-body system under their internal forces (assuming no external forces) can be studied using the Lagrangian analysis.
We assume that both the bodies experience equal and opposite forces and rotate around the common centre of mass. Then, the differential equations describing the orbits (see (1), (2) and (3) in image) have the reduced mass terms to represent combined motion of both the masses.
In these equations, if we assume that the central mass is very large compared to the planet mass, then the planet mass (m2) term vanishes and orbit equations become dependent only on the mass of the central body (m1). This situation is similar to the Schwarzschild solution results (eq. (4) in image) for a large central mass. The GR analysis suggests that the space-time around the large central mass gets curved and therefore the planets have to necessarily follow a curved path. The nature of the curved path is independent of the mass of the planet. Therefore, even a photon (which has zero mass) path has to bend as the space itself is curved.
But, we may also have a situation where the planet mass is not very small compared with the central star. Then according to the classical analysis, the rotation of both the bodies around the common centre will have to be considered. In a classical analysis, this two-body system can be represented mathematically as a single reduced mass rotating around the centre. Similarly, an analysis for the two-body system, based on GR should also consider the resultant curvature created by the both masses.
The question is: How to include the reduced mass term in the GR analysis? This is necessary to ensure that the results of the classical analysis and GR are consistent with each other.

REPLY [0 votes]: We may (?) conduct the GR analysis of two-body problem in two steps:

Find out geodesic equations (GR analysis) for the orbits of the second body (a big planet like the Jupiter), assuming that the first body (Sun) is fixed at the centre. Plot them on a graph. As per relativity, working in any frame should be allowed. In any case, an observer making measurements (on earth)  is neither on the star nor at the centre of mass frame of that planet-star combination.

Geometrically transfer (or transform) the orbits to the centre of mass frame. We can draw another graph by measuring the distance between various points on the orbit and the centre of mass using the first graph. The precession of the orbits, if any, will also get transferred to the centre of mass frame graph.

This approach can be easier, if a direct mathematical transformation/analysis of a two-body system is difficult.


TITLE: Siimplifying $(n+1)\cdot (n+1)! + (n+1)!-1$
QUESTION [2 upvotes]: I'm trying to prove something and I have stumbled accross following line in my math book:
\begin{align*}
 & = (n+1)\cdot (n+1)! + (n+1)! - 1 \\
& = [(n+1) + 1]\cdot(n+1)! - 1 \\
&
\end{align*}
But I'm unable to understand how that is possible. First, how can someone even see this relation in the first way? Second, why can we summarise (+1)⋅(+1)! in such a way?

REPLY [1 votes]: $$(n+1)(n+1)!+(n+1)!-1=(n+1+1)(n+1)!-1=(n+2)(n+1)!-1=(n+2)!-1$$


text:
\begin{document}

\title{Four Dimensional Quantum Yang-Mills Theory and Mass Gap}
\author{Simone Farinelli
        \thanks{Simone Farinelli, Ettenbergstrasse 13,
                CH-8907 Wettswil, Switzerland, e-mail simone.farinelli@alumni.ethz.ch}
        }
\maketitle

\begin{abstract}
A quantization procedure for the Yang-Mills equations for the Minkowski space $\mathbf{R}^{1,3}$ is carried out in such a way that field maps
satisfying Wightman's axioms of Constructive Quantum Field Theory can be obtained. Moreover, the spectrum of the corresponding Hamilton operator is proven to be positive and bounded away from zero except for the case of the vacuum state which has vanishing energy level. The particles corresponding to all solution fields are bosons.
\end{abstract}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\theoremstyle{definition}
\newtheorem{ex}{Example}[section]
\newtheorem{rem}{Remark}[section]
\newtheorem*{nota}{Notation}
\newtheorem{defi}{Definition}
\newtheorem{conjecture}{Conjecture}

\tableofcontents

\section{Introduction}
Yang-Mills fields, which are also called gauge fields, are used in modern physics to describe physical fields that play the role of carriers of an interaction (cf. \cite{EoM02}). Thus, the electromagnetic field in electrodynamics, the field of vector bosons, carriers of the weak interaction in the Weinberg-Salam theory of electrically weak interactions, and finally, the gluon field, the carrier of the strong interaction, are described by Yang-Mills fields. The gravitational field can also be interpreted as a Yang-Mills field (see \cite{DP75}).\par
The idea of a connection as a field was first developed by H. Weyl (1917), who also attempted to describe the electromagnetic field in terms of a connection. In 1954, C.N. Yang and R.L. Mills (cf. \cite{MY54}) suggested that the space of intrinsic degrees of freedom of elementary particles (for example, the isotropic space describing the two degrees of freedom of a nucleon that correspond to its two pure states, proton and neutron) depends on the points of space-time, and the intrinsic spaces corresponding to different points are not canonically isomorphic.\par In geometrical terms, the suggestion of Yang and Mills was that the space of intrinsic degrees of freedom is a vector bundle over space-time that does not have a canonical trivialization, and physical fields are described by cross-sections of this bundle. To describe the differential evolution equation of a field one has to define a connection in the bundle, that is, a trivialization of the bundle along the curves in the base. Such a connection with a fixed holonomy group describes a physical field, usually called a Yang-Mills field. The equations for a free Yang-Mills field can be deduced from a variational principle. They are a natural non-linear generalization of Maxwell's equations.\par
Field theory does not give the complete picture. Since the early part of the 20th century, it has been understood that the description of nature at the subatomic scale requires quantum mechanics, where classical observables correspond to typically non commuting self-adjoint operators on a Hilbert space and classic notions as ``the trajectory of a particle'' do not apply. Since fields interact with particles, it became clear by the late 1920s that an internally coherent account of nature must incorporate quantum concepts for fields as well as for particles. When doing this components of fields at different points in space-time become non-commuting operators.\par
The most important Quantum Field Theories describing elementary particle physics are gauge theories formulated in terms of a principal fibre bundle over the Minkowskian space-time with particular choices of the structure group. They are depicted in Table \ref{GT}.\par
\begin{table}[!]
\begin{center}
\begin{tabular}{|l|l|l|}
  \hline
  \textbf{Gauge Theory} & \textbf{Fundamental Forces}& \textbf{Structure Group}\\
  \hline\hline
  Quantum Electrodynamics  & Electromagnetism &  $U(1)$\\
  (QED) & & \\
  \hline
  Electroweak Theory & Electromagnetism  & $\text{SU}(2)\times U(1)$\\
  (Glashow-Salam-Weinberg) & and weak force & \\
  \hline
  Quantum Chromodynamics  & Strong force & $\text{SU}(3)$\\
  (QCD) & & \\
  \hline
  Standard Model  & Strong, weak forces    & $\text{SU}(3)\times\text{SU}(2)\times U(1)$\\
   & and electromagnetism & \\
   \hline
   Georgi-Glashow Grand   & Strong, weak forces    & $\text{SU}(5)$\\
   Unified Theory (GUT1)& and electromagnetism & \\
  \hline
   Fritzsch-Georgi-Minkowski    & Strong, weak forces    & $\text{SO}(10)$\\
   Grand Unified Theory (GUT2)& and electromagnetism & \\
  \hline
   Grand Unified  & Strong, weak forces    & $\text{SU}(8)$\\
   Theory (GUT3)& and electromagnetism & \\
  \hline
  Grand Unified  & Strong, weak forces    & $\text{O}(16)$\\
   Theory (GUT4)& and electromagnetism & \\
  \hline
\end{tabular}
\end{center}
\caption{Gauge Theories}\label{GT}
\end{table}
In order for Quantum Chromodynamics to completely explain the observed world of strong interactions, the theory must
imply:
\begin{itemize}
\item \textbf{Mass gap:} There must exist some positive constant $\Delta$ such that the excitation of the vacuum state has energy at least $\Delta$. This would explain why the nuclear force is strong but short-ranged.
\item \textbf{Quark confinement:} The physical particle states corresponding to proton, neutron and pion must be $\text{SU}(3)$-invariant. This would explain why individual quarks are never observed.
\item \textbf{Chiral symmetry breaking:} In the limit for vanishing quark-bare masses the vacuum is invariant under a certain subgroup of the full symmetry group that acts on the quark fields. This is needed to account for the ``current algebra'' theory of soft pions.
\end{itemize}

\noindent The seventh CMI-Millenium prize problem is the following conjecture.
\begin{conjecture}\label{CMI}
For any compact simple Lie group $G$ there exists a nontrivial Yang-Mills theory on the Minkowskian $\mathbf{R}^{1,3}$, whose quantization satisfies Wightman's axiomatic properties of Constructive Quantum Field Theory and has a mass gap $\eta>0$.
\end{conjecture}
The conjecture is explained in \cite{JW04} and commented in \cite{Do04} and in \cite{Fad05}. To our knowledge this conjecture is unproved. There have been serious attempts by Jormakka(\cite{Jo10}) and Dynin (\cite{Dy13}). See also the work of Frasca (\cite{Fr07}).

It is important at what stage quantization is performed. In Jormakka(\cite{Jo10}) the classical Yang-Mills equations are solved first, and the solution quantized afterwards. But Quantum Field Theory needed to prove Conjecture \ref{CMI} requires the quantization of the Yang-Mills equations first and  the solution of the quantized problem, then. Moreover, Jormakka, whose original aim was to tackle Conjecture \ref{CMI}, constructs solutions of the classical Yang-Mills equations but does not completely verify all Wightman's axioms and concludes by an argument requiring generalized eigenvectors and eigenvalues that there is no positive spectral gap. We believe this last conclusion is wrong.  Dynin provides a context for non perturbative quantum field theory in nuclear Kree-Gelfand triples, solves the Yang-Mills equation on Euclidean balls and presents the anti-normal quantization of the Yang-Mills energy-mass functional in Kree-Gelfand triples proving the existence of a positive spectral gap. Frasca proves by means of perturbation theory that in the limit of the coupling constant going to infinity a quantized Yang-Mills theory is equivalent to $\lambda\phi^4$ theory with the dynamics ruled just by a homogeneous equation. He therefore proves that there exist a spectral gap in the strong coupling limit.
\par  This paper is organized as follows. Section 2 presents the Yang-Mills equations
and their Hamiltonian formulation for the Minkowskian $\mathbf{R}^{1,3}$. Section 3 depicts Wightman's axioms of Constructive Quantum Field Theory, which are verified in Sections 4, where Yang-Mills Equations are quantized and the existence of a positive mass gap proven.
Section 5 concludes.

\section{Yang-Mills Connections}
\subsection{Definitions, Existence and Uniqueness}\label{ptrel}
A Yang-Mills connection is a connection in a principal fibre bundle over a (pseudo-)Riemannian manifold whose curvature satisfies the harmonicity condition i.e. the Yang-Mills equation.
\begin{defi}[\textbf{Yang-Mills Connection}]
 Let $P$ be a principal $G$-fibre bundle over a pseudoriemannian $m$-dimensional manifold $(M,h)$, and let $V$ be the vector bundle associated with $P$ and $\mathbf{C}^K$, induced by the representation $\rho:G\rightarrow\text{GL}(\mathbf{C}^K)$. A connection on the principal fibre bundle $P$ is Lie-algebra $\mathcal{G}$ valued one form $A$ on $M$. It defines a connection $\nabla$ for the vector bundle $V$, i.e. an operator acting on the space of cross sections of $V$. The vector bundle connection $\nabla$ can be extended to an operator $d:\Gamma(\bigwedge^p(M)\bigotimes V)\rightarrow \Gamma(\bigwedge^{p+1}(M)\bigotimes V)$, by the formula
\begin{equation}
d^{\nabla}(\eta\otimes v):= d\eta\otimes v + (-1)^{p}\eta\otimes \nabla v.
\end{equation}
The operator $\delta^{\nabla}:\Gamma(\bigwedge^{p+1}(M)\bigotimes V)\rightarrow \Gamma(\bigwedge^{p}(M)\bigotimes V)$, defined as the formal adjoint to $d$ is equal to
\begin{equation}
\delta^{\nabla}\eta= (-1)^{p+1}\ast d^{\nabla} \ast,
\end{equation}
\noindent where $\ast$ denotes the Hodge-star operator.\par
A connection $A$ in a principal fibre bundle $P$ is called a Yang-Mills field if the curvature $F:=dA+A\wedge A$, considered as a $2$-form with values in the vector bundle $\mathcal{G}$, satisfies the Yang-Mills equations
\begin{equation}\label{YME}
 \delta^{\nabla}F=0,
\end{equation}
 or, equivalently
\begin{equation}
 \delta^{\nabla}R^{\nabla}=0,
 \end{equation}
 where $R^{\nabla}(X,Y):=\nabla_X\nabla_Y-\nabla_Y\nabla_X-\nabla_{[X,Y]}$ denotes the curvature of the vector bundle $V$ and is a $2$-form with values in $V$.
\end{defi}
\begin{rem}[\textbf{Representations of Connections on Vector and Principle Fibre Bundles}]
Since the connection $A$ is a Lie algebra $\mathcal{G}$ valued $1$-form on $M$, the fields $A_j(x):=A(x)e_j=\sum_{k=1}^KA_j^k(x)t_k$ define by means of the tangential map $T_e\rho:\mathcal{G}\rightarrow\mathcal{L}(\mathbf{C}^K)$ of the representation $\rho:G\rightarrow\text{GL}(\mathbf{C}^K)$, fields of endomorphisms $T_e\rho A_1,\dots,T_e\rho A_m\in\mathcal{L}(V_x)$ for the bundle $V$. Given a basis of the Lie algebra $\mathcal{G}$ denoted by $\{t_1,\dots,t_K\}$, the endomorphisms $\{w_s:=T_e\rho.t_s\}_{s=1,\dots,K}$ in $\mathcal{L}(\mathbf{C}^K)$ have matrix representations with respect to a local basis $\{v_s(x)\}_{s=1,\dots,K}$ denoted by $[w_s]_{\{v_s(x)\}}$. Being $\rho$ a representation, $T_e\rho$ has maximal rank and the endomorphism are linearly independent. Given a local basis $\{e_j(x)\}_{j=1,\dots,m}$ for $x\in U\subset M$, the Christoffel symbols of the connection $\nabla$ are locally defined by the equation
\begin{equation}
\nabla_{e_j}v_s=\sum_{r=1}^K\Gamma_{j,s}^rv_r,
\end{equation}
holding on $U$ and satisfy the equalities
\begin{equation}
\Gamma_{j,s}^r=\sum_{a=1}^K[w_a]_s^rA^a_j.
\end{equation}
Given a local vector field $v=\sum_{s=1}^Kf^sv_s$ in $V|_U$ and a local vector field $e$ in $TM|_U$ the connection $\nabla$ has a local representation
\begin{equation}
\nabla_ev=\sum_{s=1}^K(df^s(e).v_s+f^s\omega(e).v_s),
\end{equation}
where $\omega$ is an element of $T^*U|_U\bigotimes \mathcal{L}(V|_U)$ ie. an endomorphism valued $1-$form satisfying
\begin{equation}
\omega(e_j)v_s=\sum_{r=1}^K\Gamma_{j,s}^rv_r.
\end{equation}
\end{rem}
\begin{rem}The curvature two form reads in local coordinates as
\begin{equation}
F=\sum_{1\le i<j\le M}\sum_{k=1}^KF_{i,j}^k\,t_k\,dx_i\wedge dx_j=\frac{1}{2}\sum_{i,j=1}^M\sum_{k=1}^K\left(\partial_j A^k_i-\partial_i A_j^k-\sum_{a,b=1}^KC^k_{a,b}A^a_iA_j^b\right)t_k<,dx_i\wedge dx_j,
\end{equation}
where $C=[C^c_{a,b}]_{a,b,c=1,\dots,K}$ are the \textit{structure constants} of the Lie algebra $\mathcal{G}$ corresponding to the basis $\{t_1,\dots,t_K\}$, which means that for any $a,b$
\begin{equation}
[t_a,t_b]=\sum_{c=1}^KC_{a,b}^ct_c.
\end{equation}
\end{rem}
The existence and uniqueness of solutions of the Yang-Mills equations in the Minkowski space have been first established by Segal (cf. \cite{Se78} and \cite{Se79}), who proves that the corresponding Cauchy problem encoding initial regular data has always a unique local and global regular solution. He proves as well that the temporal gauge ($A_0=0$) chosen to express the solution does not affect generality, because any solution of the Yang-Mills equation can be carried to one satisfying the temporal gauge. This subject has been undergoing intensive research, improving the original results. For example in \cite{EM82} and in \cite{EM82Bis} the Yang-Mills-Higgs equations, which generalize (\ref{YME}) and are non linear order two PDEs, have been reformulated in the temporal gauge as a non linear order one PDE satisfying a constraint equation. This PDE can be written as integral equation solving (always and uniquely, locally and globally) the Cauchy data problem with improved regularity results. Existence, uniqueness and regularity of the Yang-Mills-Higgs equations under the MIT Bag boundary conditions have been investigated in \cite{ScSn94} and \cite{ScSn95}.

\subsection{Hamiltonian Formulation for the Minkowski Space}
 The Hamilton function describes the dynamics of a physical system in classical mechanics by means of Hamilton's equations. Therefore, we have to reformulate the Yang-Mills equations in Hamiltonian mechanical terms. We focus our attention on the Minskowski $\mathbb{R}^4$ with the pseudoriemannian structure of special relativity $h=dx^0\otimes dx^0-dx^1\otimes dx^1-dx^2\otimes dx^2-dx^3\otimes dx^3$. The coordinate $x^0$ represents the time $t$, while $x^1,x^2,x^3$ are the space coordinates.\par We introduce Einstein's summation notation and adopt the convention that indices for coordinate variables from the greek alphabet vary over $\{0,1,2,3\}$ and those from the latin alphabet vary over the space indices $\{1,2,3\}$. For a generic field $F=[F_\mu]_{\mu=0,1,2,3}$ let $\mathbf{F}:=[F_i]_{i=1,2,3}$ denote the ``space'' component. The color indices lie in $\{1,\dots,K\}$. Let \begin{equation}\varepsilon^{a,b,c}:=\left\{
                                                       \begin{array}{ll}
                                                         +1 & \hbox{($\pi$ is even)} \\
                                                         -1 & \hbox{($\pi$ is odd)} \\
                                                         \;\;\; 0 & \hbox{(two indices are equal),}
                                                       \end{array}
                                                     \right.
\end{equation}and any other choice of lower and upper indices, be the Levi-Civita symbol, defined by mean of the permutation $\pi:=\left(
                              \begin{array}{ccc}
                                1 & 2 & 3 \\
                                a & b & c \\
                              \end{array}
                            \right)$ in $\mathfrak{S}^3$.
\begin{rem}
If the Lie group $G$ is simple, then the Lie-Algebra is simple and the structure constants can be written as
\begin{equation}
C_{a,b}^c = g\varepsilon^c_{a,b},
\end{equation}
for a positive constant$g$ called \textit{coupling constant}, (see f.i. \cite{We05} Chapter 15, Appendix A). The components of the curvature read then
\begin{equation}
F_{\mu,\nu}^k=\frac{1}{2}(\partial_\nu A^k_\mu-\partial_\mu A_\nu^k-g\varepsilon^k_{a,b}A^a_\mu A^b_\nu).
\end{equation}
We will consider only simple Lie groups.
\end{rem}
We need to introduce an appropriate gauge for the connections we are considering.
\begin{defi}[\textbf{Coulomb Gauge}]
A connection $A$ over the Minkowski space satisfies the \textit{Coulomb gauge} if and only if
\begin{equation}
\partial_jA_j^a=0
\end{equation}
for all $a=1,\dots,K$.
\end{defi}
\begin{defi}[\textbf{Transverse Projector}]
Let $\mathcal{F}$ be the Fourier transform on functions in $L^2(\mathbf{R}^3,\mathbf{R})$. The transverse projector $T:L^2(\mathbf{R}^3,\mathbf{R}^3)\rightarrow L^2(\mathbf{R}^3,\mathbf{R}^3)$ is defined as
\begin{equation}
(Tv)_i:=\mathcal{F}^{-1}\left(\left[\delta_{i,j}-\frac{p_ip_j}{|p|^2}\right]\mathcal{F}(v_j)\right),
\end{equation}
and the vector field $v$ decomposes as sum of a \textit{transversal} ($v^{\perp}$) and  \textit{longitudinal} ($v^{\parallel}$) component:
\begin{equation}
v_i=v_i^{\perp}+v_i^{\parallel},\quad v_i^{\perp}:=(Tv)_i,\quad v_i^{\parallel}:= v_i-(Tv)_i.
\end{equation}
\end{defi}
\begin{rem}
The Coulomb gauge condition for a connection $A$ is equivalent with the vanishing of its longitudinal component:
\begin{equation}
{A_i^a}^{\parallel}(t,\cdot)=0
\end{equation}
for all $i=1,2,3$, all $a=1,\dots K$ and any $t\in\mathbf{R}$.
\end{rem}
\begin{proposition}\label{ModGreen}
For a simple Lie group as structure group let $A$ be a connection over the Minkowskian $\mathbf{R}^4$ satisfying the Coulomb gauge and assume that $A_i^a(t,\cdot)\in C^{\infty}(\mathbf{R}^3,\mathbf{R})\cap L^2(\mathbf{R}^3,\mathbf{R})$ for all $i=1,2,3$, all $a=1,\dots K$ and any $t\in\mathbf{R}$. The operator $L$ on the real Hilbert space $L^2(\mathbf{R}^3,\mathbf{R}^K)$ defined as
\begin{equation}
L=L(\mathbf{A};x)=[L^{a,b}(\mathbf{A};x)]:=[\delta^{a,b}\Delta_x^{\mathbf{R}^3}+g\varepsilon^{a,c,b}A_k^c(t,x)\partial_k]
\end{equation}
is essentially self adjoint and elliptic for any time parameter $t\in\mathbf{R}$. Its spectrum lies on the real line and decomposes into discrete $\spec_d(L)$ and continuous spectrum $\spec_c(L)$.  If $0$ is an eigenvalue, then it has finite multiplicity, i.e. $\ker(L)$ is always finite dimensional .\par
The modified Green's function $G=G(\mathbf{A};x,y)=[G^{a,b}(\mathbf{A};x,y)]\in\mathcal{S}^{\prime}(\mathbf{R}^3,\mathbf{R}^{K\times K})$ for the operator $L$ is the distributional solution to the equation
\begin{equation}
L^{a,b}(\mathbf{A};x)G^{b,d}(\mathbf{A};x,y)=\delta^{a,d}\delta(x-y)-\sum_{n=1}^{N}\psi_n^a(x,\mathbf{A})\psi_n^d(y,\mathbf{A}),
\end{equation}
where $\{\psi_n(\mathbf{A}; \cdot)\}_n$ is an o.n. $L^2$-basis of $N$-dimensional $\ker(L)$.
This modified Green's function can be written as Riemann-Stielties integral: for any $\varphi\in\mathcal{S}(\mathbf{R}^3,\mathbf{R}^K)\cap L^2(\mathbf{R}^3,\mathbf{R}^K)$
\begin{equation}\label{gen}
G(\mathbf{A};x,\cdot)(\varphi)=\int_{\lambda\neq0}\frac{1}{\lambda}d(E_{\lambda}\varphi)(x),
\end{equation}
where $(E_{\lambda})_{\lambda\in\mathbf{R}}$ is the resolution of the identity corresponding to $L$.
\end{proposition}
\begin{rem}
In \cite{Br03} and \cite{Pe78} the modified Green's function is constructed assuming that the operator $L$ has a discrete spectral resolution $(\psi_n(\mathbf{A};\cdot),\lambda_n)_{n\ge0}$ as
\begin{equation}\label{part}
G(\mathbf{A};x,y)=\sum_{n:\lambda_n\neq0}\frac{1}{\lambda_n}\psi_n(\mathbf{A};x)\psi_n^{\dagger}(\mathbf{A};y).
\end{equation}
Since the discontinuity points of the spectral resolution $(E_{\lambda})_{\lambda\in\mathbf{R}}$ are the eigenvalues, i. e. the elements of $\spec_d(L)$ (cf. \cite{Ri85}, Chapter 9), the solution (\ref{gen}) extends (\ref{part}) to the general case.
\end{rem}
\begin{rem}
Rigged Hilbert spaces have been introduced in mathematical physics to utilize Dirac calculus for the spectral theory of operators appearing in quantum mechanics (see \cite{Ro66}, \cite{ScTw98} and \cite{Ma08}). Within that framework the role of distributions to provide a rigorous foundation to generalized eigenvectors and eigenvalues is highlighted by the Gel'fand-Kostyuchenko spectral theorem (see \cite{Ze09} Chapter 12.2.4 and \cite{GS64} Section I.4).
\begin{theorem}[\textbf{Gel'fand-Kostyuchenko}]\label{GK} Let $A:\mathcal{S}(\mathbf{R}^N)\rightarrow\mathcal{S}(\mathbf{R}^N)$ be a linear sequentially continuous operator, which is essentially selfadjoint in the Hilbert Space $L^2(\mathbf{R}^N)$. Then, the operator $A$ has a complete system $(F_m)_{m\in M}$ of eigendistributions. This means that there exists a non empty index set $M$ such that $F_M\in\mathcal{S}^{\prime}(\mathbf{R}^N)$ for all $m\in M$ and
\begin{enumerate}
\item[(i)] Eigendistributions: there exist a function $\lambda:M\rightarrow\mathbf{R}$ such that
\begin{equation}
F_m(A\varphi)=\lambda(m)F_m(\varphi),
\end{equation}
for all $m\in M$ and all test functions $\varphi\in\mathcal{S}(\mathbf{R}^N)$.
\item[(ii)] Completeness: if $F_m(\varphi)=0$ for all $m\in M$ and fixed test function $\varphi\in\mathcal{S}(\mathbf{R}^N)$, then $\varphi=0$.
\end{enumerate}
The spectrum of $A$ reads then $\spec(A)=\lambda(M)$, where for all $m\in M$ the real number $\lambda(m)$ is either an eigenvalue (i.e. $F_m\in L^2(\mathbf{R}^N)$) or an element of the continuous spectrum (i.e. $F_m\notin L^2(\mathbf{R}^N)$).
\end{theorem}

\end{rem}

\begin{proof}[Proof of Proposition \ref{ModGreen}]
As long as the connection satisfies the Coulomb gauge condition, the operator $L$ is symmetric and essentially selfadjoint on the appropriate domain, as a direct computation involving integration by parts can show. Being its leading symbol elliptic, the operator $L$ is elliptic and restricted to $[-\frac{R}{2}, +\frac{R}{2}]^3$ under the Dirichlet boundary conditions it has a discrete spectral resolution (cf. \cite{Gi95}, Chapter 1.11.3). Every eigenvalue has finite multiplicity. The dimension of the eigenspaces is an integer valued continuous and hence constant function of $R$. For $R\rightarrow+\infty$ the discrete Dirichlet spectrum of $L$ on $[-\frac{R}{2}, +\frac{R}{2}]^3$ clusters in the spectrum of $L$ on $\mathbf{R}^3$ which decomposes into discrete and continuous spectrum. Therefore, the eigenvalues must have finite multiplicity and, in particular $\ker(L)$ is finite dimensional.\\
Equation (\ref{gen}) gives the modified Green's function as it can be verified by the following computation which holds true for any $\varphi\in\mathcal{S}(\mathbf{R}^3,\mathbf{R}^K)\cap L^2(\mathbf{R}^3,\mathbf{R}^K)$:
\begin{equation}
\begin{split}
L(\mathbf{A};x)G(\mathbf{A};x,\cdot)(\varphi)&=L\int_{\lambda\neq0}\frac{1}{\lambda}d(E_{\lambda}\varphi)(x)=\int_{\lambda\neq0}\frac{1}{\lambda}Ld(E_{\lambda}\varphi)(x)=\\
&=\int_{\mathbf{R}}d(E_{\lambda}\varphi)(x)-\int_{0^-}^{0^+}d(E_{\lambda}\varphi)(x)=\phi(x)-P_{\ker(L)}\varphi(x)=\\
&=\delta(x-\cdot)(\varphi)-\sum_{n=1}^{N}\psi_n(x,\mathbf{A})\psi_n(\cdot,\mathbf{A})(\varphi).
\end{split}
\end{equation}
\end{proof}
The existence of eigenvalues of $L$ depends on the additive perturbation to the Laplacian given by $g\varepsilon^{a,c,b}A_k^c(t,x)\partial_k$. For example, if $g=0$ or $\mathbf{A}=0$, the operator $L$ has no eigenvalues and $\spec(L)=\spec_c(L)=]-\infty,0]$. In general the spectrum depends on the choice of the connection $A$. In \cite{BEP78} and in \cite{Pe78} special cases comprising pure gauges and Wu-Yang monopoles are computed explicitly. We are interested in a reformulation of the general solution (\ref{gen}), where the dependence on the connection becomes explicit. Inspired by \cite{Ha97} we find
\begin{proposition}\label{prop}
For a simple Lie group as structure group, if we assume that the coupling constant $g<1$, then a Green's function $K=K(\mathbf{A};x,y)=[K^{a,b}(\mathbf{A};x,y)]$ for the operator $L$ in Proposition \ref{ModGreen}, that is, a distributional solution to the equation
\begin{equation}
L^{a,b}(\mathbf{A};x)K^{b,d}(\mathbf{A};x,y)=\delta^{a,d}\delta(x-y),
\end{equation}
is given by the convergent series in $\mathcal{S}^{\prime}(\mathbf{R}^3,\mathbf{R}^{K\times K})$
\begin{equation}\label{ser}
\begin{split}
K^{b,d}(\mathbf{A};x,y)=&\frac{\delta^{b,d}}{4\pi|x-y|}+g\varepsilon^{d,e,b}\int_{\mathbf{R}^3}d^3u_1\frac{1}{4\pi|x-u_1|}A_k^e(u_1)\partial_k\left(\frac{1}{4\pi|u_1-y|}\right)+\\
&+\dots+\\
&+(-1)^{n-1}\frac{n+1}{2}g^n\varepsilon^{b,e_1,s_1}\cdots\varepsilon^{s_{n-1},e_n,d}\int_{\mathbf{R}^3}d^3u_1\frac{1}{4\pi|x-u_1|}A_k^{e_1}(u_1)\cdot\\
&\cdot\partial_k\int_{\mathbf{R}^3}d^3u_2\frac{1}{4\pi|u_1-u_2|}\dots\int_{\mathbf{R}^3}d^3u_n\frac{1}{4\pi|u_{n-1}-u_n|}A_l^{e_n}(u_n)\partial_l\left(\frac{1}{4\pi|u_n-y|}\right)+\\
&+\dots
\end{split}
\end{equation}
\end{proposition}
\begin{proof}
The series (\ref{ser}) converges because of the integrability of the connection $A$ and the fact that $g<1$. Recall that $\frac{1}{|x|}\in L^1_{loc}(\mathbf{R}^3)\subset\mathcal{S}^{\prime}(\mathbf{R}^3)$. We now check that it represents a Green's function for $L$:
\begin{equation}
L(\mathbf{A};x)^{a,b}K^{b,d}(\mathbf{A};x,y)=\delta^{a,d}\delta(x-y)+\lim_{n\rightarrow+\infty}\text{Rest}_n,
\end{equation}
where, after having evaluated a ``telescopic sum'', the remainder part reads
\begin{equation}
\begin{split}
\text{Rest}_n=&(-1)^{n-1}\frac{n+1}{2}g^{n+1}\varepsilon^{b,e_1,s_1}\cdots\varepsilon^{s_{n-1},e_n,d}\varepsilon^{a,c,b}A^c_k(x)\int_{\mathbf{R}^3}d^3u_1\frac{-x_k}{4\pi|x-u_1|^3}A_k^{e_1}(u_1)\cdot\\
&\cdot\partial_k\int_{\mathbf{R}^3}d^3u_2\frac{1}{4\pi|u_1-u_2|}\dots\int_{\mathbf{R}^3}d^3u_n\frac{1}{4\pi|u_{n-1}-u_n|}A_l^{e_n}(u_n)\partial_l\left(\frac{1}{4\pi|u_n-y|}\right).
\end{split}
\end{equation}
Because of the integrability of the connection, there exists a constant $C>0$ such that for any $\varphi\in\mathcal{S}(\mathbf{R}^3,\mathbf{R})$
\begin{equation}
|\text{Rest}_n(\varphi)|\le Cg^{n+1}\|\varphi\|_{L^2(\mathbf{R}^3,\mathbf{R})}\rightarrow0\quad(n\rightarrow+\infty),
\end{equation}
and the proposition follows.
\end{proof}
\begin{rem}
For the Minkowskian $\mathbf{R}^4$ the assumption of a (dimensionless) coupling constant $g<1$ is well posed, as the experimental results on running coupling constants show: for the Yang-Mills theory for QCD with SU(3) one is basically allowed to choose any value $g>0$ (see \cite{SaSc10}). Moreover, we will later have to analyze the case where $g\rightarrow0^+$.
\end{rem}
\begin{corollary}\label{cor}
Under the same assumptions as Proposition \ref{prop} the distribution
\begin{equation}
G(\mathbf{A};x,y)= K(\mathbf{A};x,y)+K(\mathbf{A};x,y)^{\dagger}-\sum_{n:\lambda_n=0}\psi_n(\mathbf{A};x)\psi_n^{\dagger}(\mathbf{A};y)
\end{equation}
is a (symmetric) modified Green's function for the operator $L$.
\end{corollary}
\noindent After this preparation we can turn to the Hamiltonian formulation of Yang-Mills' equations, following the results in \cite{Br03} and \cite{Pe78}, which just need to be adapted for the generic case that $L$ has a mixture of discrete and continuous spectral resolution.
\begin{theorem}\label{CHam}. For a simple Lie group as structure group and for canonical variables  satisfying the Coulomb gauge condition, the Yang-Mills equations for the Minkowskian $\mathbf{R}^4$  can be written as Hamilton equations
\begin{equation}
\left\{
  \begin{array}{ll}
    \frac{d\mathbf{E}}{dt}&=-\frac{\partial H}{\partial \mathbf{A}}(\mathbf{A},\mathbf{E})\\
    \\
    \frac{d\mathbf{A}}{dt}&=+\frac{\partial H}{\partial \mathbf{E}}(\mathbf{A},\mathbf{E})
  \end{array}
\right.
\end{equation}
for the following choices:
\begin{itemize}
\item \textbf{Position variable:} $\mathbf{A}=[A_{i}^a(t,x)]_{\substack{ a=1,\dots,K \\ i=1,2,3}}$ also termed {\it potentials} ,
\item \textbf{Momentum variable:} $\mathbf{E}=[E_{i}^a(t,x)]_{\substack{ a=1,\dots,K \\ i=1,2,3}}$ termed {\it chromoelectric fields},
\item \textbf{Hamilton Function:} defined as function of $\mathbf{A}$ and $\mathbf{E}$ as
 \begin{equation}\label{classicH}
  H=H(\mathbf{A},\mathbf{E}):=\frac{1}{2}\int_{\mathbf{R}^3}d^3x\left(E_i^a(t,x)^2+B_i^a(t,x)^2+f^a(t,x))\Delta f^a(t,x)+2\rho^c(t,x)A^c_0(t,x)\right),
 \end{equation}
 \noindent where $\mathbf{B}=[B_i^a]$, termed {\it chromomagnetic fields} is the matrix valued function defined as
 \begin{equation}
     \begin{split}
       B_i^a&:=\frac{1}{4}\varepsilon_{i}^{j,k}\left(\partial_jA_k^a-\partial_kA^a_j+g\varepsilon^a_{b,c}A_j^bA_k^c\right),
     \end{split}
 \end{equation}
\noindent and $\rho=[\rho^a(t,x)]$, termed {\it charge density} is  the vector valued function defined as
\begin{equation}
\rho^a:=g\varepsilon^{a,b,c}E^b_iA^c_i,
\end{equation}
\noindent and where
\begin{equation}\label{deff}
\begin{split}
f^a(t,x)&:=-g\int_{\mathbf{R}^3}d^3y\,G^{a,b}(\mathbf{A};x,y)\rho^b(t,x)\\
A_0^a(t,x)&:=\int_{\mathbf{R}^3}d^3y\,G^{a,b}(\mathbf{A};x,y)\Delta f^b(t,x),
\end{split}
\end{equation}
\noindent for the modified Green's function $G(\mathbf{A};x,y)$ for the operator $L(\mathbf{A};x)$.
\end{itemize}
\end{theorem}
\begin{rem}
As shown in \cite{Pe78} the ambiguities discussed by Gribov in \cite{Gr78} concerning the gauge fixing (see also \cite{Si78} and \cite{He97}) can be traced precisely to the existence of zero eigenfunctions of the operator $L$.
\end{rem}

\section{Wightman's Axioms of Constructive Quantum Field Theory}
In year 1956 Wightman first stated the axioms needed for CQFT in his seminal work \cite{Wig56}, which remained not very widely spread in the scientific community till 1964, when the first edition of \cite{SW10} appeared. We will list the axioms in the slight refinement of \cite{BLOT89} and \cite{DD10}.
\begin{defi}[\textbf{Wightman's Axioms}]
A scalar (respectively vectorial-spinorial) quantum field theory consists of a separable Hilbert space $\mathcal{H}$, whose elements are called states, a unitary representation $U$ of the Poincar\'{e} group $\mathcal{P}$ in $\mathcal{H}$, an operator valued distributions $\Phi$ (respectively $\Phi_1,\dots,\Phi_d$) on $\mathcal{S}(\mathbf{R}^4)$ with values in the unbounded operators of $\mathcal{H}$, and a dense subspace $\mathcal{D}\subset\mathcal{H}$ such that the following properties hold:
\begin{description}
  \item[(W1) Relativistic invariance of states:] The representation $U:\mathcal{P}\rightarrow\mathcal{U}(\mathcal{H})$ is strongly continuous.
  \item[(W2) Spectral condition:] Let $P_0,P_1,P_2,P_3$ be the infinitesimal generators of the one-parameter groups $t\mapsto U(te_\mu,I)$ for $\mu=0,1,2,3$. The operators $P_0$ and $P_0^2-P_1^2-P_2^2-P_3^2$ are positive. This is equivalent to the spectral measure $E_{\cdot}$ on $\mathbf{R}^4$ corresponding to the restricted representation $\mathbf{R}^4\ni a\mapsto U(a,I)$ having support in the positive light cone (cf. \cite{RS75}, Chapter IX.8).
  \item[(W3) Existence and uniqueness of the vacuum:] There exists a unique state $\psi_0\in\mathcal{D\subset\mathcal{H}}$ such that $U(a,I)\psi_0=\psi_0$ for all $a\in\mathbf{R}^4$.
  \item[(W4) Invariant domains for fields:] The maps $\Phi:\mathcal{S}(\mathbf{R}^4)\rightarrow\mathcal{O}(\mathcal{H})$, and, respectively $\Phi_1,\dots,\Phi_d:\mathcal{S}(\mathbf{R}^4)\rightarrow\mathcal{O}(\mathcal{H})$, from the Schwartz space of test function to (possibly) unbounded selfadjoint operators on the Hilbert space satisfies (satisfy) following properties
      \begin{description}
      \item[(a)] For all $\varphi\in\mathcal{S}(\mathbf{R}^4)$ and all field maps the domain of definitions $\mathcal{D}(\Phi(\varphi))$, $\mathcal{D}(\Phi(\varphi)^*)$, and respectively $\mathcal{D}(\Phi_j(\varphi))$, $\mathcal{D}(\Phi_j(\varphi)^*)$, all contain $\mathcal{D}$ and the restrictions of all operators to $\mathcal{D}$ agree.
      \item[(b)] $\Phi(\varphi)(\mathcal{D})\subset\mathcal{D}$, and, respectively $\Phi_j(\varphi)(\mathcal{D})\subset\mathcal{D}$.
      \item[(c)] For any $\psi\in\mathcal{D}$ fixed the maps $\varphi\mapsto\Phi(\varphi)\psi$, and, respectively $\varphi\mapsto\Phi_j(\varphi)\psi$, are linear.
      \end{description}
  \item[(W5) Regularity of fields:] For all $\psi_1,\psi_2\in\mathcal{D}$ the map $\varphi\mapsto(\psi_1,\Phi(\varphi)\psi_2)$, and, respectively the maps $\varphi\mapsto(\psi_1,\Phi_j(\varphi)\psi_2)$, are tempered distributions i.e. elements of $\mathcal{S}^{\prime}(\mathbf{R}^4)$.
  \item[(W6) Poincar\'{e} invariance:] For all $(a,\Lambda)\in\mathcal{P}$, $\varphi\in\mathcal{S}(\mathbf{R}^4)$ and $\psi\in\mathcal{D}$ the inclusion $U(a,\Lambda)\mathcal{D}\subset\mathcal{D}$ must hold and
  \begin{description}
  \item[(Scalar field case): ] The following equation must hold for all $\Lambda\in \text{O}(1,3)$
  \begin{equation}
  U(a,\Lambda)\Phi(\varphi)U(a,\Lambda)^{-1}\psi=\Phi((a,\Lambda)\varphi)\psi.
  \end{equation}
  \item[(Vectorial/Spinorial field case): ] There exists a representation of $\text{SL}(2,\mathbf{C})$ on $\mathbf{C}^d$ denoted by $\rho$ and satisfying $\rho(-I)=I$ or $\rho(I)=I$ such that for all $\Lambda\in \text{SO}^+(1,3)$ and $\Phi:=[\Phi_1,\dots,\Phi_d]^{\dagger}$
  \begin{equation}
  U(a,\Lambda)\Phi(\varphi)U(a,\Lambda)^{-1}\psi=\rho(s^{-1}(\Lambda))\Phi((a,\Lambda)\varphi)\psi,
  \end{equation}
  where $s$ denotes the spinor map $s:\text{SL}(2,\mathbf{C})\rightarrow\text{SO}^{+}(1,3)$ defined as following.
  The vector spaces of Hermitian matrices $\mathfrak{H}$ in $\mathbf{C}^{2\time 2}$ and $\mathbf{R}^4$ are isomorphically mapped  by
    \begin{equation}
     \begin{split}
      X:\mathbf{R}^4&\longrightarrow \mathfrak{H}\\
      x=(x^0, x^1, x^2, x^3)&\mapsto X(x):=\left[
                                          \begin{array}{cc}
                                            x^0+x^3 & x^1-ix^2 \\
                                            x^1+ix^2 & x^0-x^3 \\
                                          \end{array}
                                        \right].
      \end{split}
    \end{equation}
    The group $\text{SL}(2,\mathbf{C})$ acts on $\mathfrak{H}$ by
    \begin{equation}
     \begin{split}
      \text{SL}(2,\mathbf{C})\times\mathfrak{H} &\longrightarrow \mathfrak{H}\\
      (P,X)&\mapsto P.X:=PXP^*.
      \end{split}
    \end{equation}
    The spinor map is defined as
    \begin{equation}\label{spinormap}
     \begin{split}
      s: \text{SL}(2,\mathbf{C})&\longrightarrow \text{SO}^+(1,3) \\
        P&\mapsto s(P):x\mapsto s(P)x:=X^{-1}(PX(x)P^*).
      \end{split}
    \end{equation}
  \end{description}
  \item[(W7) Microscopic casuality or local commutativity:] Let $\varphi,\chi\in\mathcal{S}(\mathbf{R}^4)$, whose supports are spacelike separated, i.e. $\phi(x)\chi(y)=0$ if $x-y$ is not in the positive light cone. Then,
      \begin{description}
      \item[(Scalar field case): ] The images of the test functions by the map field must commute
          \begin{equation}
          [\Phi(\varphi), \Phi(\chi)]=0.
          \end{equation}
      \item[(Vectorial/Spinorial field case): ] For any field maps $j,i=1,\dots,d$ either the commutations
          \begin{equation}
          [\Phi_j(\varphi), \Phi_i(\chi)]=0,\quad [\Phi_j^*(\varphi), \Phi_i(\chi)]=0,
          \end{equation}
          or the anticommutations
          \begin{equation}
          [\Phi_j(\varphi), \Phi_i(\chi)]_{+}=0,\quad[\Phi_j^*(\varphi), \Phi_i(\chi)]_{+}=0
          \end{equation}
          hold.
      \end{description}
  \item[(W8) Cyclicity of the vacuum:] The sets
  \begin{description}
      \item[(Scalar field case): ] The set
     \begin{equation}
        \mathcal{D}_0:=\{\Phi(\varphi_1)\cdots\Phi(\varphi_n)\psi_0\,|\,\varphi_j\in\mathcal{S}(\mathbf{R}^4), n\in\mathbf{N}_0\}
     \end{equation}\noindent is dense in $\mathcal{H}$.
      \item[(Vectorial/Spinorial field case): ]  The set \begin{equation}\mathcal{D}_0:=\{\Psi(\varphi_1)\cdots\Psi(\varphi_n)\psi_0\,|\,\varphi_j\in\mathcal{S}(\mathbf{R}^4), \Psi\in\{\Phi_1,\dots,\Phi_d,\Phi_1^*,\dots,\Phi_d^*\},\,n\in\mathbf{N}_0\}\end{equation}\noindent is dense in $\mathcal{H}$.
      \end{description}
\end{description}

\end{defi}
We remark that Osterwalder and Schrader (cf. \cite{OS73}, \cite{OS73Bis}) utilized the Wick rotation technique to pass from the Minkowskian to the Euclidean space and formulate axioms equivalent to Wightman's ones in terms of Euclidean Green's functions. In \cite{OS75} they defined free Bose and Fermi fields and proved a Feynman-Ka\v{c} formula for boson-fermion models. For a general overview see also \cite{GJ81}.\par
The dynamics of the quantized system obtained by Wightman's axioms is given by the Schr\"odinger equation
\begin{equation}
\imath\frac{\partial}{\partial x^0}\psi=H\psi,
\end{equation}
where $\psi\in\mathcal{H}$ and $H$ is the unbounded, selfadjoint Hamilton operator, obtained by a quantization procedure of the Hamilton function in the classical description of the physical system.\par

\subsection{Second Quantization}
To extend a quantum mechanical model accounting for a fixed number of particles to one accounting for an arbitrary number, the following  procedure is required.

\begin{defi}[\textbf{Second Quantization}]
Let $\mathcal{H}$ be the Hilbert space whose unit sphere corresponds to the possible pure quantum states of the system with a fixed number of particles. The
\textbf{Fock space} $\mathcal{F}(\mathcal{H}):= \bigoplus_{n=0}^\infty\mathcal{H}^{(n)}$ where, $\mathcal{H}^{(0)}:=\mathbf{C}$ and $\mathcal{H}^{(n)}:=\mathcal{H}\otimes\cdots\mathcal{H}$ ($n$ times tensor product) is the vector space representing the states of a quantum system with a variable number of particles. The vector $\Omega_0:=(1,0\dots)\in\mathcal{F}(\mathcal{H})$ is called the \textbf{vacuum vector}. Given $\psi\in\mathcal{F}(\mathcal{H})$, we write $\psi^{(n)}$ for the orthogonal projection of $\psi$ onto $\mathcal{H}^{(n)}$. The set $F_0$ consisting of those $\psi$ such that $\psi^{(n)}=0$ for all sufficiently large $n$ is a dense subspace of the Fock space, called the \textbf{space of finite particles}. The \textbf{symmetrization} and  {anti-symmetrization} operators
\begin{equation}
    \begin{split}
     S_n(\psi_1\otimes\dots\otimes\psi_n)&:=\frac{1}{n!}\sum_{\sigma\in\mathfrak{S}^n}\psi_{\sigma(1)}\otimes\dots\otimes\sigma_{\sigma(n)}\\
     A_n(\psi_1\otimes\dots\otimes\psi_n)&:=\frac{1}{n!}\sum_{\sigma\in\mathfrak{S}^n}(-1)^{\text{sgn}(\sigma)}\psi_{\sigma(1)}\otimes\dots\otimes\sigma_{\sigma(n)}
    \end{split}
\end{equation}
extend by linearity to $\mathcal{H}^{(n)}$ and are projections. The state space for $n$ fermions is defined as $\mathcal{H}^{(n)}_a:=A_n(\mathcal{H}^{n})$ and that for $n$ bosons as $\mathcal{H}^{(n)}_s:=S_n(\mathcal{H}^{n})$. The \textbf{fermionic Fock space} is defined as
\begin{equation}
\mathcal{F}_a(\mathcal{H}):=\bigoplus_{n=0}^{\infty}\mathcal{H}^{(n)}_a,
\end{equation}
and the \textbf{bosonic Fock space} as
\begin{equation}
\mathcal{F}_s(\mathcal{H}):=\bigoplus_{n=0}^{\infty}\mathcal{H}^{(n)}_s.
\end{equation}
A \textit{unitary} operator $U:\mathcal{H}\rightarrow\mathcal{H}$ can be uniquely extended to a \textit{unitary} operator $\Gamma(U):\mathcal{F}(\mathcal{H})\rightarrow\mathcal{F}(\mathcal{H})$ as
\begin{equation}
\Gamma(U)|_{\mathcal{H}^{(n)}}:=\bigotimes_{j=1}^n U.
\end{equation}
 A \textit{selfadjoint} operator $A$ on $\mathcal{H}$ with dense subspace $\mathcal{D}(A)\subset\mathcal{H}$ can be uniquely extended to a \textit{selfadjoint} operator $d\Gamma(A)$ on $\mathcal{F}(\mathcal{H})$ as closure of the essentialy selfadjoint operator
\begin{equation}
d\Gamma(A)|_{\mathcal{D}(d\Gamma A)\cap\mathcal{H}^{(n)}}:=\bigoplus_{j=1}^n\mathbb{1}\otimes\dots\mathbb{1}\otimes\underbrace{A}_{j}\otimes\mathbb{1}\dots\mathbb{1},
\end{equation}
where
\begin{equation}
\mathcal{D}(d\Gamma(A)):=\left\{\psi\in F_0\left|\,\psi^{(n)}\in\bigotimes_{j=1}^n\mathcal{D}(A)\text{ for each }n\right.\right\}.
\end{equation}
The operator $d\Gamma(A)$ is called the \textbf{second quantization} of $A$.
\end{defi}
It is easy to prove that the the spectrum of the second quantization can be infered from the spectrum of the first.
\begin{proposition}\label{spec}
Let $A$ be a selfadjoint operator with a discrete spectral resolution, i.e. $A\varphi_j=\lambda_j\varphi_j$, where $\{\lambda_j\}_{j\ge0}\subset\mathbb{R}$ and $\{\varphi_j\}_{j\ge0}$ is a o.n.B in $\mathcal{H}$. Then, $d\Gamma(A)|_{\mathcal{H}^{(n)}}$ has a discrete spectral resolution given by
\begin{equation}
d\Gamma(A)|_{\mathcal{H}^{(n)}}\varphi_{i_1}\otimes\dots\otimes\varphi_{i_n}=\left(\sum_{j=1}^n\lambda_{i_j}\right)\varphi_{i_1}\otimes\dots\otimes\varphi_{i_n}\quad(i_j\ge0, 1\le j\le n).
\end{equation}
If  $A$ is selfadjoint operator with continuous spectrum $\spec_c(A)$, then $d\Gamma(A)|_{\mathcal{H}^{(n)}}$ has a continuous spectrum given by
\begin{equation}
\spec_c\left(d\Gamma(A)|_{\mathcal{H}^{(n)}}\right)=\left\{\left.\sum_{j=1}^n\lambda_j\right|\,\lambda_j\in\spec_c(A)\text{ for }1\le j\le n\right\}.
\end{equation}
\end{proposition}
\subsection{Segal Quantization}
\begin{defi}[\textbf{Segal Quantization}]
Let $f\in\mathcal{H}$ be fixed. For vectors in $\mathcal{H}^{(n)}$ of the form $\eta=\psi_1\otimes\psi_2\otimes\dots\otimes\psi_n$ we define a map $b^{-}(f):\mathcal{H}^{(n)}\rightarrow \mathcal{H}^{(n-1)}$ by
\begin{equation}
b^{-}(f)\eta:=(f,\psi_1)\psi_2\otimes\dots\otimes\psi_n.
\end{equation}
\noindent The expression $b^{-}(f)$ extends by linearity to a bounded operator on $\mathcal{F}(\mathcal{H})$. The operator $N:=d\Gamma(I)$ is termed \textbf{number operator} and
\begin{equation}
a^{-}(f):=\sqrt{N+1}b^{-}(f)
\end{equation}
the \textbf{annihilation operator} on $\mathcal{F}_s(\mathcal{H})$. Its adjoint, $a^{-}(f)^*$ is called the \textbf{creation operator}. Finally, the real linear (but not complex linear) operator
\begin{equation}
\Phi_S(f):=\frac{1}{\sqrt{2}}\left(a^{-}(f)+a^{-}(f)^*\right)
\end{equation}
is termed \textbf{Segal field operator}, and the map
\begin{equation}
\begin{split}
\Phi_S:\mathcal{H}&\rightarrow\mathcal{F}_s(\mathcal{H})\\
f&\mapsto\Phi_S(f)
\end{split}
\end{equation}
the \textbf{Segal quantization} over $\mathcal{H}$.
\end{defi}
\begin{theorem}\label{Segal}
Let $\mathcal{H}$ be a complex Hilbert space and $\Phi_S$ the corresponding Segal quantization. Then:
\begin{description}
\item[(a)] For each $f\in\mathcal{H}$ the operator $\Phi_S(f)$ is essentially selfadjoint on $F_0$.
\item[(b)] The vacuum $\Omega_0$ is in the domain of all finite products $\Phi_S(f_1)\Phi_S(f_2)\dots\Phi_S(f_n)$ and the linear span of $\left\{\Phi_S(f_1)\Phi_S(f_2)\dots\Phi_S(f_n)\Omega_0\,|\,f_i\in\mathcal{H},n\ge0\right\}$ is dense in $\mathcal{F}_s(\mathcal{H})$.
\item[(c)] For each $\psi_0\in F_0$ and $f,g\in\mathcal{H}$
    \begin{equation}
    \begin{split}
     &\Phi_S(f)\Phi_S(g)\psi-\Phi_S(g)\Phi_S(f)\psi=\imath\text{Im}(f,g)\psi\\
     &\exp\left(\imath\Phi_S(f+g)\right)=\exp\left(-\frac{\imath}{2}\text{Im}(f,g)\right)\exp\left(\imath\Phi_S(f)\right)\exp\left(\imath\Phi_S(g)\right).
    \end{split}
    \end{equation}

\item[(d)]If $f_n\rightarrow f$ in $\mathcal{H}$, then:
    \begin{equation}
      \begin{split}
      \Phi_S(f_n)\rightarrow \Phi_S(f)\\
      \exp\left(\imath\Phi_S(f_n)\right)\rightarrow \exp\left(\imath\Phi_S(f)\right)\\
      \end{split}
    \end{equation}
\item[(e)]For every unitary operator $U$ on $\mathcal{H}$, $\Gamma(U):\mathcal{D}(\overline{\Phi_S(f)})\rightarrow\mathcal{D}(\overline{\Phi_S(Uf)})$ and for $\psi\in\mathcal{D}(\overline{\Phi_S(Uf)})$ and for all $f\in\mathcal{H}$
    \begin{equation}
    \Gamma(U)\overline{\Phi_S(f)}\Gamma(U)^{-1}\psi=\overline{\Phi_S(Uf)}\psi.
    \end{equation}
\end{description}
\end{theorem}
\begin{proof} See Theorem X.41 in \cite{RS75}.
\end{proof}

\section{Quantization of Yang-Mills Equations and Positive Mass Gap}
\subsection{Functional Derivation, Functional Integration and Hilbert Space Construction}
In this subsection we follow the work of Kuo (\cite{Ku96}) which is aligned with the framework for functional integration developed
by Cartier/DeWitt-Morette (see \cite{Ca97}, \cite{La04} and \cite{CDM10}). We briefly summarize the construction without proofs.\par
Given the space of tempered distributions $\mathcal{S}^{\prime}(\mathbf{R}^N)$, the dual of the space of test functions $\mathcal{S}(\mathbf{R}^N)$ we have the following Gel'fand triple:
\begin{equation}\label{Gelf1}
\mathcal{S}(\mathbf{R}^N)\subset L^2(\mathbf{R}^N)\subset\mathcal{S}^{\prime}(\mathbf{R}^N),
\end{equation}
where $\mathcal{S}(\mathbf{R}^N)$ and $\mathcal{S}^{\prime}(\mathbf{R}^N)$ are Banach spaces and  $L^2(\mathbf{R}^N)$ an Hilbert space. This Gel'fand triple is backed by the continuous inclusions
\begin{equation}
\mathcal{S}(\mathbf{R}^N)\subset L^{s,2}(\mathbf{R}^N)\subset L^2(\mathbf{R}^N)\subset L^{-s,2}(\mathbf{R}^N))\subset\mathcal{S}^{\prime}(\mathbf{R}^N),
\end{equation}
where $L^{s,2}(\mathbf{R}^N)$ denotes the Sobolev space of distributions such that all weak derivatives up to order $s$ are square integrable.\par By Minlos theorem there exist a unique probability measure $\mu$  such that
\begin{equation}
\int_{\mathcal{S}^{\prime}(\mathbf{R}^N)}d\mu(F)\,e^{\imath F(\varphi)}=e^{-\frac{1}{2}\|\varphi\|^2_{L^2(\mathbf{R}^N)}}\quad\text{ for all } \varphi\in\mathcal{S}(\mathbf{R}^N).
\end{equation}
\noindent The probability space $(\mathcal{S}^{\prime}(\mathbf{R}^N),\mu)$ is termed \textit{white noise space} and the measure $\mu$ is called the standard \textit{Gaussian measure} on $\mathcal{S}^{\prime}(\mathbf{R}^N)$.\par
The construction of the square integrable functions can be mimicked in infinite dimensions for the integration domain, where the white noise space $(\mathcal{S}^{\prime}(\mathbf{R}^N),\mu)$ plays the role of its finite dimensional analogue $(\mathbf{R}^N,dx^N)$. We obtain the space of the square integrable functionals over the white noise space $L^2(\mathcal{S}^{\prime}(\mathbf{R}^N),\mu)$. Furthermore,  following the \textit{Kubo-Takenaka construction} explained in Chapter 4.2 of \cite{Ku96}, which utilizes the Wiener-it\^{o} theorem and the second quantization of the partial differential operator $\Pi_{i=1}^N[-\partial_i^2+x_i^2+1]$, it is possible to obtain a nuclear space $\mathcal{E}(\mathcal{S}^{\prime}(\mathbf{R}^N),\mu)$, \textit{the space of the Hida test functions} and its dual $\mathcal{E}^{\prime}(\mathcal{S}^{\prime}(\mathbf{R}^N),\mu)$, \textit{the space of the Hida distributions}, and the Gelf'and triple:
\begin{equation}
\mathcal{E}(\mathcal{S}^{\prime}(\mathbf{R}^N),\mu)\subset L^2(\mathcal{S}^{\prime}(\mathbf{R}^N),\mu)\subset \mathcal{E}^{\prime}(\mathcal{S}^{\prime}(\mathbf{R}^N),\mu).
\end{equation}
More exactly, the Kubo-Takenaka construction produces the following continuous inclusions for all $s>0$:
\begin{equation}
\mathcal{E}(\mathcal{S}^{\prime}(\mathbf{R}^N),\mu)\subset L^{s,2}(\mathcal{S}^{\prime}(\mathbf{R}^N),\mu)\subset L^2(\mathcal{S}^{\prime}(\mathbf{R}^N),\mu)\subset L^{-s,2}(\mathcal{S}^{\prime}(\mathbf{R}^N),\mu)\subset \mathcal{E}^{\prime}(\mathcal{S}^{\prime}(\mathbf{R}^N),\mu),
\end{equation}
 and $L^{s,2}(\mathcal{S}^{\prime}(\mathbf{R}^N),\mu)$ the Hida-Sobolev space of test functions.\par
We can conclude that the Feynman integral can be properly defined for functionals on the white noise space with a standard Gaussian measure:
\begin{equation}
\int_{\mathcal{S}^{\prime}(\mathbf{R}^N)}\mathcal{D}(F)\,\varphi(F):=\int_{\mathcal{S}^{\prime}(\mathbf{R}^N)}d\mu(F)\,\varphi(F),
\end{equation}
and that it can be formally defined for functionals in the space of Hida distribution as the value of such distributions on the Hida test function $1$:
\begin{equation}
\int_{\mathcal{S}^{\prime}(\mathbf{R}^N)}\mathcal{D}(F)\,\Phi(F):=\int_{\mathcal{S}^{\prime}(\mathbf{R}^N)}\mathcal{D}(F)\,\Phi(F)\cdot 1:=\Phi(1).
\end{equation}
\begin{defi}[\textbf{Functional Derivative}] Let $\varphi\in L^2(\mathcal{S}^{\prime}(\mathbf{R}^N),\mu)$. Its functional derivative is defined as G\^{a}teaux derivative as
\begin{equation}
\frac{\delta\varphi}{\delta F}(F).G:=\left.\frac{d}{d\epsilon}\right|_{\epsilon=0}\varphi(F+\epsilon G),
\end{equation}
for any $G\in\mathcal{S}^{\prime}(\mathbf{R}^N)$.
\end{defi}
\begin{rem}
The Gel'fand-Kostyuchenko spectral theorem (Theorem \ref{GK}) can be lifted to an analogous version for selfadjoint operators on the Hilbert space $L^2(\mathcal{S}^{\prime}(\mathbf{R}^N),\mu)$.
\end{rem}

\subsection{Quantization}
To account for functionals on transversal fields as required by the Coulomb gauge, we perform the Kubo-Takenaka construction for
\begin{equation}
L^2_{\bot}(\mathbf{R}^4,\mathbf{R}^{K \times 3}):=\{\left.\mathbf{A}\in L^2(\mathbf{R}^4,\mathbf{R}^{K \times 3})\right|\,\mathbf{A}^{\parallel}=0\},
\end{equation}
obtaining the Gel'fand triples
\begin{equation}
\begin{split}
&\mathcal{S}_{\bot}(\mathbf{R}^4,\mathbf{R}^{K \times 3})\subset L^2_{\bot}(\mathbf{R}^4,\mathbf{R}^{K \times 3})\subset\mathcal{S}^{\prime}_{\bot}(\mathbf{R}^4,\mathbf{R}^{K \times 3})\\
&\mathcal{E}(\mathcal{S}^{\prime}_{\bot}(\mathbf{R}^4,\mathbf{R}^{K \times 3}),\mu)\subset L^2(\mathcal{S}^{\prime}_{\bot}(\mathbf{R}^4,\mathbf{R}^{K \times 3}),\mu)\subset \mathcal{E}^{\prime}(\mathcal{S}^{\prime}_{\bot}(\mathbf{R}^4,\mathbf{R}^{K \times 3}),\mu).
\end{split}
\end{equation}
We introduce the Hamilton operator originated by the quantization of the Hamiltonian formulation of Yang-Mills equations,
and verify that the Wightman's axioms are satisfied. Quantizing Thorem \ref{CHam} we obtain
\begin{theorem}\label{quantization}
Let $H=H(A,E)$ be the Hamilton function of the Hamilton equations equivalent to the Yang-Mills equations as in Theorem \ref{CHam} for a simple Lie group as structure group. The quantization of the position variable $A$, of the momentum variable $E$ and of the Hamilton function means the following substitution:
\begin{equation}
\begin{split}
\mathbf{A} \in C^{\infty}(\mathbf{R}^4,\mathbf{R}^{K\times 3})&\longrightarrow \mathbf{A}\in\mathcal{O}(L^2(\mathcal{S}^{\prime}_{\bot}(\mathbf{R}^4,\mathbf{R}^{K \times 3}),\mu))\\
\mathbf{E} \in C^{\infty}(\mathbf{R}^4,\mathbf{R}^{K\times 3})&\longrightarrow \frac{1}{\imath}\frac{\delta}{\delta \mathbf{A}}\in\mathcal{O}(L^2(\mathcal{S}^{\prime}_{\bot}(\mathbf{R}^4,\mathbf{R}^{K \times 3}),\mu))\\
H \in C^{\infty}(\mathbf{R}^{K\times 3}\times\mathbf{R}^{K\times 3},\mathbf{R})&\longrightarrow H\left(\mathbf{A},\frac{1}{\imath}\frac{\delta}{\delta \mathbf{A}}\right)\in\mathcal{O}(L^2(\mathcal{S}^{\prime}_{\bot}(\mathbf{R}^4,\mathbf{R}^{K \times 3}),\mu)).
\end{split}
\end{equation}
The Hamilton operator for the quantized Yang-Mills equations, denoted again by $H$ reads
\begin{equation}
H=H_I+H_{II}+H_{III},
\end{equation}
where
\begin{equation}
\begin{split}
H_I&=-\frac{1}{2}\int_{\mathbf{R}^3}d^3x\left[\frac{\delta}{\delta A_i^a(t,x)}\frac{\delta}{\delta A_i^a(t,x)}\right]\\
& \\
H_{II}&=\frac{1}{16}\int_{\mathbf{R}^3}d^3x\left[\varepsilon_i^{j,k}\varepsilon_i^{p,q}(\partial_jA^a_k-\partial_kA_j^a+g\varepsilon^{a,b,c}A_j^bA_k^c)(\partial_pA^a_q-\partial_qA_p^a+g\varepsilon^{a,b,c}A_p^bA_q^c)\right]\\
& \\
H_{III}&=-\frac{g^2}{2}\int_{\mathbf{R}^3}d^3x\int_{\mathbf{R}^3}d^3y\int_{\mathbf{R}^3}d^3\bar{y}\left[\partial_iG^{a,b}(\mathbf{A};x,y)\varepsilon^{b,c,d}A_k^d(t,y)\frac{\delta}{\delta A_k^c(t,y)}\right]\\
 &\qquad\qquad\qquad\qquad\qquad\qquad\quad\;\left[\partial_iG^{a,b}(\mathbf{A};x,\bar{y})\varepsilon^{b,c,d}A_k^d(t,\bar{y})\frac{\delta}{\delta A_k^c(t,\bar{y})}\right].
\end{split}
\end{equation}
With the domain of definition
\begin{equation}
\mathcal{D}(H):=\left\{\Psi\in L^2(\mathcal{S}^{\prime}_{\bot}(\mathbf{R}^4,\mathbf{R}^{K \times 3}),\mu)\left|\,H\Psi\in L^2(\mathcal{S}^{\prime}_{\bot}(\mathbf{R}^4,\mathbf{R}^{K \times 3},\mu)\right.\right\}
\end{equation}
the operator $H$  is selfadjoint.
\end{theorem}
\begin{rem} The operator $H_I$ is the Laplace operator in infinite dimensions for functionals of the potential fields. The operator $H_II$ is a multiplication operator corresponding to the fibrewise multiplication with the square of the connection curvature. Both operators $H_I$ and $H_{II}$ do not vanish if $g=0$ and do not contribute to the existence of a mass gap. The operator $H_{III}$ vanishes if $g=0$ and -as we will see- is responsible for the existence of a mass gap.
\end{rem}
\begin{proof}[Proof of Theorem \ref{quantization}]
The expressions for the operators $H_I$ and $H_{II}$ are obtained by a straightforward calculation, after having inserted the quantization in the first two addenda of (\ref{classicH}). For the operator $H_{III}$ some more work is needed. Inserting (\ref{deff}) in the last addendum of (\ref{classicH})
\begin{equation}\label{classicH3}
\begin{split}
&\int_{\mathbf{R}^3}d^3x\left(\frac{1}{2}f^a(t,x)\Delta f^a(t,x)+\rho_c(t,x)A_0^c(t,x)\right)=\\
&=\int_{\mathbf{R}^3}d^3x\left[\frac{1}{2}\left(-\int_{\mathbf{R}^3}d^3y\,G^{a,b}(\mathbf{A};x,y)\rho_b(t,y)\right)\left(-\int_{\mathbf{R}^3}d^3\bar{y}\,G^{a,b}(\mathbf{A};x,\bar{y})\rho_b(t,\bar{y})\right)\right.+\\
&\qquad\qquad\quad+\left.\rho_c(t,x)\int_{\mathbf{R}^3}d^3y\,G^{c,b}(\mathbf{A};x,y)\int_{\mathbf{R}^3}d^3\bar{y}\,\Delta G^{b,d}(\mathbf{A};y,\bar{y})\rho_d(t,\bar{y})\right]=\\
&=\int_{\mathbf{R}^3}d^3x\int_{\mathbf{R}^3}d^3y\int_{\mathbf{R}^3}d^3\bar{y}\left[\frac{1}{2}\left(G^{a,b}(\mathbf{A};x,y)\rho_b(t,y)\Delta G^{a,d}(\mathbf{A};x,\bar{y})\rho_d(t,\bar{y})\right)\right.+\\
&\qquad\qquad\quad-\left.\rho_c(t,x)G^{c,b}(\mathbf{A};x,y)\Delta G^{b,d}(\mathbf{A};y,\bar{y})\rho_d(t,\bar{y})\right]=\\
&=\int_{\mathbf{R}^3}d^3x\int_{\mathbf{R}^3}d^3y\int_{\mathbf{R}^3}d^3\bar{y}\left[\frac{1}{2}\left(G^{a,b}(\mathbf{A};x,y)\Delta G^{a,d}(\mathbf{A};x,\bar{y})\rho_b(t,y)\rho_d(t,\bar{y})\right)\right.+\\
&\qquad\qquad\quad-\left.G^{a,b}(\mathbf{A};y,x)\Delta G^{a,d}(\mathbf{A};x,\bar{y})\rho_b(t,y)\rho_d(t,\bar{y})\right]=\\
&=\frac{1}{2}\int_{\mathbf{R}^3}d^3x\int_{\mathbf{R}^3}d^3y\int_{\mathbf{R}^3}d^3\bar{y}\left[\partial_iG^{a,b}(\mathbf{A};x,y)\rho_b(t,y)\right]\left[\partial_iG^{a,d}(\mathbf{A};x,\bar{y})\rho_d(t,\bar{y})\right],
\end{split}
\end{equation}
where in the last formula transformation we have utilized integration by parts in the variables $x_1, x_2, x_3$ and the fact that $G(\mathbf{A};x,y)=G(\mathbf{A};y,x)$. Introducing the quantization into the charge we obtain
\begin{equation}
\rho_b(t,x)=-\frac{g}{\imath}\varepsilon^{b,c,d}A_k^c(t,x)\frac{\delta}{\delta A_k^c(t,x)},
\end{equation}
which inserted into \ref{classicH3} leads to
\begin{equation}
\begin{split}
H_{III}&=-\frac{g^2}{2}\int_{\mathbf{R}^3}d^3x\int_{\mathbf{R}^3}d^3y\int_{\mathbf{R}^3}d^3\bar{y}\left[\partial_iG^{a,b}(\mathbf{A};x,y)\varepsilon^{b,c,d}A_k^d(t,y)\frac{\delta}{\delta A_k^c(t,y)}\right]\\
 &\qquad\qquad\qquad\qquad\qquad\qquad\quad\left[\partial_iG^{a,b}(\mathbf{A};x,\bar{y})\varepsilon^{b,c,d}A_k^d(t,\bar{y})\frac{\delta}{\delta A_k^c(t,\bar{y})}\right].
\end{split}
\end{equation}
The proof is completed.
\end{proof}
\begin{rem}
In the physical literature (see f.i. \cite{Sch08}) the operator $C=C(\mathbf{A};x,y)$
\begin{equation}
C^{a,b}(\mathbf{A};x,y):=-\int_{\mathbf{R}^3}d^3\bar{y}G^{a,c}(\mathbf{A};x\bar{y})\Delta G^{c,b}(\mathbf{A};y,\bar{y})
\end{equation}
is termed \textit{Coulomb operator} and $G=G(\mathbf{A};x,y)$ is also called the \textit{Faddeev-Popov operator}.
\end{rem}
\begin{corollary}\label{corspec}
The spectra of $H_I$, $H_{II}$ and $H_{III}$ are:
\begin{equation}\label{specH}
\begin{split}
&\spec(H_I)=\spec_c(H_I)=[0,+\infty[\\
&\spec(H_{II})=\spec_c(H_{II})=[0,+\infty[\\
&\spec(H_{III})=\spec_c(H_{III})=\{0\}\cup[\eta,+\infty[, \text{ for a }\eta>0.
\end{split}
\end{equation}
Moreover $\eta=O(g^2)$.
\end{corollary}
\begin{proof}
We will construct generalized eigenvectors to show that the spectra have only a continuous part depicted as in (\ref{specH}). First, we analyze the operator $H_I$, which can be seen as
\begin{equation}
H_I=-\frac{1}{2}\int_{\mathbf{R}^3}d^3 x\Delta_{\mathbf{A}(t,x)}.
\end{equation} Let  $x\in\mathbf{R}^3$ and $t\in\mathbf{R}$ now be fixed. For any $R>0$ the Laplace operator $\Delta_{\mathbf{A}}$ on $[-\frac{R}{2},+\frac{R}{2}]^{3K}$ under Dirichlet boundary conditions has a discrete spectral resolution $(\lambda_k,\psi_k)_{k\ge0}$, where $\lambda_k=-\frac{\pi^2}{R^2}(k+1)$ and $\psi_j=\psi_j(\mathbf{A})\in C^{\infty}_0([-\frac{R}{2},+\frac{R}{2}]^{3K},\mathbf{C})$. We can extend $\psi_j$ outside the cube by setting its value to $0$ obtaining an approximated eigenvector for the approximated eigenvalue $\lambda_j$, which is in line with the fact that the Laplacian on $L^2(\mathbf{R}^{3K},\mathbf{C})$ has solely a continuous spectrum, which is $]-\infty,0]$. The functional
\begin{equation}
\Psi^{x_0}_k(\mathbf{A}):=\delta(\cdot-x_0)\psi_k(\mathbf{A}(t,x_0))
\end{equation}
is a generalized eigenvector in $\mathcal{E}^{\prime}(\mathcal{S}^{\prime}_{\bot}(\mathbf{R}^4,\mathbf{R}^{K \times 3}),\mu)$ for the operator $H_I$ on the rigged Hilbert space $L^2(\mathcal{S}^{\prime}_{\bot}(\mathbf{R}^4,\mathbf{R}^{K \times 3}),\mu)$ for the generalized eigenvalue $\frac{\pi^2}{R^2}(k+1)$, which, by Theorem \ref{GK}, is an element of the continuous spectrum of the non negative operator $H_I$. By varying the generalized eigenvalue over $j$ and $R$, the claim about the spectrum follows.\par
Next, we analyze the operator $H_{II}$, which can be seen as
\begin{equation}
H_{II}=\frac{1}{2}\int_{\mathbf{R}^3}d^3 x|R^{\nabla^\mathbf{A}}(t,x)|^2,
\end{equation}
where $R^{\nabla^\mathbf{A}}$ is the curvature operator associated to the connection $\mathbf{A}$. Let  $x\in\mathbf{R}^3$ now be fixed. Any non zero $\psi\in L^2(\mathbf{R}^{3K},\mathbf{C})$ is eigenvector of the multiplication with the non negative real $|R^{\nabla^\mathbf{A}}(t,x)|^2$. The functional
\begin{equation}
\Psi^{x_0}(\mathbf{A}):=\delta(\cdot-x_0)\psi(\mathbf{A}(t,x_0))
\end{equation}
is a generalized eigenvector in $\mathcal{E}^{\prime}(\mathcal{S}^{\prime}_{\bot}(\mathbf{R}^4,\mathbf{R}^{K \times 3}),\mu)$ for the operator $H_{II}$ on the rigged Hilbert space $L^2(\mathcal{S}^{\prime}_{\bot}(\mathbf{R}^4,\mathbf{R}^{K \times 3}),\mu)$ for the generalized eigenvalue $|R^{\nabla^A}(t,x_0)|^2$, which, by Theorem \ref{GK}, is an element of the continuous spectrum of the non negative operator $H_{II}$. By varying the generalized eigenvalue over $A$, and taking into account that
\begin{equation}
\inf_{\mathbf{A}\in\mathcal{A}} |R^{\nabla^\mathbf{A}}(t,x_0)|^2=0,
\end{equation}
the claim about the spectrum follows.\par
Finally, we analyze the operator $H_{III}$, which we write as
\begin{equation}
H_{III}=-\frac{g^2}{2}\int_{\mathbf{R}^3}d^3x\int_{\mathbf{R}^3}d^3y\int_{\mathbf{R}^3}d^3\bar{y}\,D_i^a(\mathbf{A};x,y)D_i^a(\mathbf{A};x,\bar{y}),
\end{equation}
\noindent
for the operator $D=D(\mathbf{A;x,y})$ defined as
\begin{equation}
D_i^a(\mathbf{A};x,y):=\partial_iG^{a,b}(\mathbf{A};x,y)\varepsilon^{b,c,d}A_k^d(t,y)\frac{\delta}{\delta A_k^c(t,y)}.
\end{equation}
Let  $x$ and $y\in\mathbf{R}^3$ now be fixed. For any $R>0$ the  operator $D_i^a(\mathbf{A};x,y)$ on $[-\frac{R}{2},+\frac{R}{2}]^{3K}$ under periodic boundary conditions has a discrete spectral resolution $(\mu_{i,k}^a(x,y),\psi_{i,k}^a(\mathbf{A};x,y))_{k\ge0}$, where
\begin{equation}
\begin{split}
\mu_{i,k}^a(x,y)&=\frac{2\pi\imath k}{\int_{-\frac{R}{2}}^{+\frac{R}{2}}dA_j^c(t,y)[\partial_iG^{a,b}(A;x,y)\varepsilon^{b,c,d}A_j^d(t,y)]^{-1}}\\
\psi_{i,k}^a(A;x,y)&=C_{i,k}^a\exp\left(\mu_{i,k}^a(x,y)\int_0^{A_j^c(t,y)}d\bar{A}_j^c(t,y)[\partial_iG^{a,b(\bar{\mathbf{A}};x,y)}\varepsilon^{b,c,d}\bar{A}_j^d(t,y)]^{-1}\right),
\end{split}
\end{equation}
where $C_{i,k}^a\in\mathbf{C}$ is a constant. We can extend $\psi_{i,k}^a(\cdot;x,y)$ outside the cube by setting its value to $0$ obtaining an approximated eigenvector for the approximated eigenvalue $\mu_{i,k}^a(x,y)$ for the operator $D_i^a(\mathbf{A};x,y)$ on $L^2(\mathbf{R}^{3K}, \mathbf{C})$, which means that $\mu_{i,k}^a(x,y)\in\spec_c(D_i^a(\mathbf{A};x,y))$. For fixed $i$ and $a$ the functional
\begin{equation}
\Psi^{a,x_0,y_0}_{i,k}(\mathbf{A}):=\delta(\cdot-x_0)\delta(y-y_0)\delta(\bar{y}-y_0)\psi_{i,k}^a(\mathbf{A};x_0,y_0)\psi_{i,k}^a(\mathbf{A};x_0,y_0)
\end{equation}
is a generalized eigenvector in $\mathcal{E}^{\prime}(\mathcal{S}^{\prime}_{\bot}(\mathbf{R}^4,\mathbf{R}^{K \times 3}),\mu)$ for the operator
\begin{equation}
H_{i,III}^a:=-\frac{g^2}{2}\int_{\mathbf{R}^3}d^3x\int_{\mathbf{R}^3}d^3y\int_{\mathbf{R}^3}d^3\bar{y}\,D_i^a(\mathbf{A};x,y)D_i^a(\mathbf{A};x,\bar{y})
\end{equation}
on the rigged Hilbert space $L^2(\mathcal{S}^{\prime}_{\bot}(\mathbf{R}^4,\mathbf{R}^{K \times 3}),\mu)$ for the generalized eigenvalue
\begin{equation}
\lambda_{i,k}^a(x_0,y_0)=\frac{2\pi^2g^2k^2}{\left[\int_{-\frac{R}{2}}^{+\frac{R}{2}}dA_j^c(t,y_0)[\partial_iG^{a,b}(\mathbf{A};x_0,y_0)\varepsilon^{b,c,d}A_j^d(t,y_0)]^{-1}\right]^2},
\end{equation}which, by Theorem \ref{GK} is an element of the continuous spectrum of the operator $H_{i,III}^a$. Since, by Proposition \ref{prop} and Corollary \ref{cor}
\begin{equation}
[\partial_iG^{a,b}(\mathbf{A};x_0,y_0)\varepsilon^{b,c,d}A_j^d(t,y_0)]^{-1}=O((1+|\mathbf{A}(t,y_0)|^2)^{-1})\qquad (|\mathbf{A}(t,y_0)|\rightarrow+\infty),
\end{equation}
the denominator of the generalized eigenvalue has a uniform bound in the connection $\mathbf{A}$ given by
\begin{equation}
\left[\int_{-\frac{R}{2}}^{+\frac{R}{2}}dA_j^c(t,y_0)[\partial_iG^{a,b}(A;x_0,y_0)\varepsilon^{b,c,d}A_j^d(t,y_0)]^{-1}\right]^2\le \left[C\int_{-\infty}^{\infty}d\alpha\,\frac{1}{1+\alpha^2}\right]^2=\pi^2 C^2.
\end{equation}
By varying the generalized eigenvalue over $k$ and $R$, the claim about the spectrum follows for $\eta:=\frac{6K g^2}{C^2}$. Note that $0$ is an element of the continuous spectrum but not of the discrete spectrum, which is empty. The proof is complete.
\end{proof}
\subsection{Main Theorem}
We describe the construction of field maps and verify Wightman's axioms.
\begin{defi}\textbf{Quantum Field Settting for Yang-Mills Theory}\label{QFTYM}
\begin{description}
\item [Pseudoriemannian manifold:] $M:=\mathbf{R}^{1,3}$ with the Minkowski metric tensor $g:=dx^0\otimes dx^0-dx^1\otimes dx^1-dx^2\otimes dx^2-dx^3\otimes dx^3$.
\item [Positive light cone:] $\mathcal{C}_+:=\{x\in M\,|\,g(x,x)\ge0\}$.
\item [Bundles over the manifold:] $P$ is a principle fibre bundle over $M$ with simple structure group $G$.
\item [Hilbert space:]$\mathcal{H}:=L^2(\mathcal{S}^{\prime}_{\bot}(\mathbf{R}^4,\mathbf{R}^{K \times 4}),\mu)$.
\item [Bosonic Fock space:] $\mathcal{F}_s(\mathcal{H})$.
\item [Vacuum state:] $\Omega_0:\equiv (1,0,0,\dots)\in\mathcal{F}_s(\mathcal{H})$.
\item [Dense subspace:] The space of test functionals $\mathcal{D}:=\mathcal{E}(\mathcal{S}^{\prime}_{\bot}(\mathbf{R}^4,\mathbf{R}^{K \times 4}),\mu)$.
\item[Unitary representation of the Poincar\'{e} group:]
 \begin{equation}
  U(a,\Lambda)\Psi(A):=\Psi(A(\Lambda\cdot+a)).
\end{equation} It is defined on any $\Psi\in\mathcal{H}$ and extended to $\mathcal{F}_s(\mathcal{H})$ by $\Gamma(U)$.
\item[Hamilton operator: ] $H$ defined in Theorem (\ref{quantization}), with domain of definition $\mathcal{D}(H)$. The definition on the Hilbert space $\mathcal{H}$ is extended to $\mathcal{F}_s(\mathcal{H})$ by $d\Gamma(H)$. Note that a functional $\Psi=\Psi(A)=\Psi([A_0;\mathbf{A}])\in L^2(\mathcal{S}^{\prime}_{\bot}(\mathbf{R}^4,\mathbf{R}^{K \times 4}),\mu)$ can be seen as an element of $L^2(\mathcal{S}^{\prime}_{\bot}(\mathbf{R}^4,\mathbf{R}^{K \times 3}),\mu)$ if we consider only the dependence on the space components $\mathbf{A}$.

\item[Field maps:]  Let us consider a connection $A$ for the principal fibre bundle $P$. Since the connection $A$ is a Lie algebra $\mathcal{G}$ valued $1$-form on $M$, we can write
\begin{equation}
A(x)=\sum_{\mu=0}^3A_\mu(x)dx^\mu
\end{equation}
for the fields $A_{\mu}(x):=A(y)e_\mu=\sum_{a=1}^KA_\mu^a(x)t_a$, where $\{t_1,\dots,t_K\}$ is a basis of $\mathcal{G}$.
   We define now following operators for $\mu=0,1,2,3$ and $a=1,\dots, K$:
   \begin{equation}
   \begin{split}
   E_\mu^a:\mathcal{S}(\mathbb{R}^4)&\rightarrow\mathcal{H}\\
   \varphi&\mapsto E_\mu^a(\varphi),\text{ defined as }E_\mu^a(\varphi)(A):=\int_{\mathbf{R}^4}d^4x\,\varphi(x)A_\mu^a(x)
   \end{split}
   \end{equation}
  and for $\varphi\in\mathcal{S}(\mathbb{R}^4)$
  \begin{equation}
  \Phi_\mu^a(\varphi):=\Phi_S(\text{Re}(E_\mu^a(\varphi)))+\imath\Phi_S(\text{Im}(E_\mu^a(\varphi))),
  \end{equation}
  where $\Phi_S$ is the closure of the Segal operator.
\end{description}
\end{defi}
\begin{rem}The field maps are well defined. As a matter of fact, for any indices $\mu=0,1,2,3$, $k=1,\dots,K$ and all $\varphi\in\mathcal{S}(\mathbf{R}^4)$ the functional $E_\mu^a(\varphi)$ is an element of $\mathcal{H}$:
\begin{equation}
\begin{split}
\int_{\mathcal{S}^{\prime}_{\bot}(\mathbf{R}^4,\mathbf{R}^{K \times 4})}\mathcal{D}(A)|E_\mu^a(\varphi)(A)|^2&=\int_{\mathcal{S}^{\prime}_{\bot}(\mathbf{R}^4,\mathbf{R}^{K \times 4})}\mathcal{D}(A)\left|\int_{\mathbf{R}^4}d^4x\,\varphi(x)A_\mu^a(x)\right|^2=\\
&\le \int_{\mathcal{S}^{\prime}_{\bot}(\mathbf{R}^4,\mathbf{R}^{K \times 4})}\mathcal{D}(A)\int_{\mathbf{R}^4}d^4x\,|\varphi(x)|^2|A_\mu^a(x)|^2=\\
&\le \|\varphi\|^2_{L^2(\mathbf{R}^4,\mathbf{C})}\int_{\mathcal{S}^{\prime}_{\bot}(\mathbf{R}^4,\mathbf{R}^{K \times 4})}\mathcal{D}(A)\|A_\mu^a\|^2_{L^2(\mathbf{R}^4,\mathbf{C})}<+\infty,
\end{split}
\end{equation}
because $\mu$ is a probability measure and, hence, $\mu\left(\mathcal{S}^{\prime}_{\bot}(\mathbf{R}^4,\mathbf{R}^{K \times 4})\right)=1$.
\end{rem}
\begin{theorem}\label{CMIThm}
The construction of Definition \ref{QFTYM} satisfies Wightman's axioms (W1)-(W8) and the spectra of the Hamilton operator $H$ contains $0$ as simple eigenvalue for the vacuum eigenstate. There exist a constant $\eta>0$ such that $\text{spec}(H)=\{0\}\cup[\eta,+\infty[$. Moreover $\eta=O(g^2)$.
\end{theorem}
\begin{proof} First we prove that Wightman's axioms are satisfied.
\begin{description}
  \item(W1): With the definition $U(a,\Lambda)\Psi(A):=\Psi(A(\Lambda\cdot+a))$ the strong continuity follows by Lebesgue's dominated convergence for the Feynman's integral and the equality
      \begin{equation}
      \|U(a,\Lambda)\Psi-U(a^{\prime},\Lambda^{\prime})\Psi\|_{L^2}^2=\int_{\mathcal{S}^{\prime}_{\bot}(\mathbf{R}^4,\mathbf{R}^{K \times 4})}\mathcal{D}(A)|\Psi(A(\Lambda\cdot+a))-\Psi(A(\Lambda^{\prime}\cdot+a^{\prime}))|^2,
      \end{equation} which holds for all $\Lambda, \Lambda^{\prime}\in\text{SO}^+(1,3)$ and all $a, a^{\prime}\in\mathbf{R}_+\times\mathbf{R}^3$.
  \item (W2): The computation of the infinitesimal generators shows that $P_\mu=\frac{1}{\imath}\frac{\partial}{\partial x\mu}$:
 \begin{equation}
P_\mu\Psi(A):=\frac{1}{\imath}\left.\frac{d}{dt}\right|_{t=0}U(te_\mu,I)\Psi(A)=\frac{1}{\imath}\left.\frac{d}{dt}\right|_{t=0}\Psi(A(\cdot+te_\mu))=\frac{1}{\imath}\frac{\partial \Psi\circ A}{\partial x_\mu}=\frac{1}{\imath}\frac{\delta \Psi}{\delta A_k}\frac{\partial A_k}{\partial X_\mu}
\end{equation}The operator $\Delta:=P_0^2-P_1^2-P_2^2-P_3^2$ is the Laplacian for the Minkowskian metric and is positive in the light cone. $P_0$ is unitary equivalent to the multiplication operator $x_0$ which is positive on $\mathcal{C}_+$. Therefore, the spectral measure in $\mathbf{R}^4$ corresponding to the restricted representation $\mathbf{R}^4\ni a\mapsto U(a,I)$ has support in the positive light cone.
  \item (W3): The vacuum state  satisfies for all $a,x\in\mathbf{R}^4$
      \begin{equation}
      \Gamma(U(a,I))\Omega_0(A)=\Omega_0(A\cdot+a)\equiv\Omega_0\equiv(1,0,\dots).
      \end{equation}
  \item (W4): follows from Theorem \ref{Segal} (a)
  \item (W5): follows from Theorem \ref{Segal} (d).
  \item (W6): For all $(a,\Lambda)\in\mathcal{P}$ the inclusion $U(a,\Lambda)\mathcal{D}\subset\mathcal{D}$  holds, because the test functional space is invariant under affine transformation of the domain. Since, by Theorem \ref{Segal} (e) for any $\mu=0,1,2,3$, $k=1,\dots,K$, $(a,\Lambda)\in\mathcal{P}$, $\varphi\in\mathcal{S}(\mathbf{R}^4)$ and $\Psi\in\mathcal{D}$
      \begin{equation}\label{tr}
      \Gamma(U(a,\Lambda))\Phi_\mu^k(\varphi)\Gamma(U(a,\Lambda))^{-1}\Psi=\Phi_\mu^k(U(a,\Lambda).\varphi)\Psi,
      \end{equation}
      which can be rewritten as
      \begin{equation}\label{tr2}
      \Gamma(U(a,\Lambda))\Phi(\varphi)\Gamma(U(a,\Lambda))^{-1}\Psi=\rho(s^{-1}(\Lambda))\Phi(U(a,\Lambda).\varphi)\Psi,
      \end{equation}
      where $s$ denotes the spinor map $s:\text{SL}(2,\mathbf{C})\rightarrow\text{SO}^{+}(1,3)$,
      \begin{equation}
       \Phi:=[\Phi_0^1,\dots,\Phi_0^K,\Phi_1^1,\dots,\Phi_1^K,\Phi_2^1,\dots,\Phi_2^K,\Phi_3^1,\dots,\Phi_3^K]^{\dagger},
      \end{equation}
      and
       \begin{equation}
      \rho\equiv\mathbb{1}_{\text{GL}(\mathbf{C}^d)}
      \end{equation}
      for $d:=4K$ is a representation of $\text{SL}(2,\mathbf{C})$ on $\mathbf{C}^d$. In particular, it satisfies $\rho(-\mathbb{1})=\mathbb{1}$, which means that representation has integer spin, as expected in the bosonic case.

  \item (W7): follows from Theorem \ref{Segal} (c). More exactly, for two test functions $\varphi$ and $\chi$ and any indices $a,b\in\{1,\dots,K\}$, $\mu,\nu\in\{0,1,2,3\}$ the commutator of the field maps can be computed as
\begin{equation}
\begin{split}
&[\Phi_\mu^a(\varphi), \Phi_\nu^b(\chi)]=\\
&=[\Phi_S(\text{Re}(E_\mu^a(\varphi))), \Phi_S(\text{Re}(E_\nu^b(\chi)))]-[\Phi_S(\text{Im}(E_\mu^a(\varphi))), \Phi_S(\text{Im}(E_\nu^b(\chi)))]+\\
&\imath\left\{[\Phi_S(\text{Re}(E_\mu^a(\varphi))), \Phi_S(\text{Im}(E_\nu^b(\chi)))]+[\Phi_S(\text{Im}(E_\mu^a(\varphi))), \Phi_S(\text{Re}(E_\nu^b(\chi)))]\right\}=\\
&=\imath\left\{(\text{Re}(E_\mu^a(\varphi)), \text{Re}(E_\nu^b(\chi)))_{L^2}-(\text{Im}(E_\mu^a(\varphi)), \text{Im}(E_\nu^b(\chi)))_{L^2}\right\}+\\
&-\left\{(\text{Re}(E_\mu^a(\varphi)), \text{Im}(E_\nu^b(\chi)))_{L^2}+(\text{Im}(E_\mu^a(\varphi)), \text{Re}(E_\nu^b(\chi)))_{L^2}\right\}.
\end{split}
\end{equation}
All the scalar products vanish because the supports of the test functions are space-like separated and hence disjoint. For example:
\begin{equation}
\begin{split}
&(\text{Re}(E_\mu^a(\varphi)), \text{Re}(E_\nu^b(\chi)))_{L^2(\mathcal{A},\mathbf{C})}=\\
&=\int_{\mathcal{S}^{\prime}_{\bot}(\mathbf{R}^4,\mathbf{R}^{K \times 4})}\mathcal{D}(A)\text{Re}\left[\int_{\mathbf{R}^4}d^4x\,\varphi(x)A^a_\mu(x)\right]\text{Re}\left[\int_{\mathbf{R}^4}d^4y\,\chi(y)A^b_\nu(y)\right]=\\
&=\int_{\mathcal{S}^{\prime}_{\bot}(\mathbf{R}^4,\mathbf{R}^{K \times 4})}\mathcal{D}(A)\int_{\mathbf{R}^4}d^4x\int_{\mathbf{R}^4}d^4y\,\text{Re}(\varphi(x))\text{Re}(\chi(y))A^a_\mu(x)A^b_\nu(y)=0.
\end{split}
\end{equation}
By Theorem \ref{Segal} (a) the field map $\Phi_\mu^a(\varphi)$ is selfadjoint and thus
\begin{equation}
[\Phi_\mu^a(\varphi)^*, \Phi_\nu^b(\chi)]=[\Phi_\mu^a(\varphi), \Phi_\nu^b(\chi)]=0.
\end{equation}
  \item (W8): follows from Theorem \ref{Segal} (b).

\end{description}

Finally, we prove the existence of a positive lower spectral bound. It suffices to prove the existence of a bound $\eta>0$ for the spectrum of  $H$ on $\mathcal{H}$. From Corollary \ref{corspec} we infer the existence of a $\eta=O(g^2)>0$ such that
\begin{equation}
\spec(H)=\spec_c(H)=\{0\}\cup[\eta,+\infty[
\end{equation}
for the Hamilton operator $H$ on the Hilbert space $\mathcal{H}$. For the spectrum of $d\Gamma(H)$ on the Fock space $\mathcal{F}(\mathcal{H})$  we obtain by Proposition \ref{spec}
\begin{equation}
\text{spec}(d\Gamma(H))= \{0\}\cup[\eta,+\infty[,
\end{equation}
where we have to consider that $0$ is a proper eigenvalue for the vacuum state $\Omega_0$.\\
The proof is completed.\\
\end{proof}

\begin{rem}
The representation $\rho$ has  integer spin and lead therefore to connection fields $(A_\mu)_{\mu=0,\dots 3}$ satisfying the Bose statistics, in accordance with the standard spin-statistics relation for gluons which are bosons.
\end{rem}

\section{Conclusion}
We have quantized Yang-Mills equations for the positive light cone in the Minkowskian $\mathbf{R}^{1,3}$ obtaining field maps
satisfying Wightman's axioms of Constructive Quantum Field Theory. Moreover, the spectrum of the corresponding Hamilton operator is positive and bounded away from zero except for the case of the vacuum state which has vanishing energy level.

\section*{Acknowledgement}
We would like to express our gratitude to Gian-Michele Graf and to Lee Smolin for the challenging discussion leading to various important corrections of the first versions of this paper.

config: arxiv


TITLE: Given a one-t0-one function f that maps M onto an arbitrary set A, prove there is a unique way to make A a manifold s.t. f becomes a diffeomorphism.
QUESTION [0 upvotes]: I'm really unsure of how to proceed, I've drawn a picture and can understand the general setting but don't know how to actually prove it.

REPLY [0 votes]: This is not possible if $f$ is not a bijection. This is because, whatever manifold structure $A$ has, a diffeomorphism is, first and foremost, a bijection.
If $f$ IS a bijection though, first define the topology on $A$ by defining a subset $U\subset A$ to be open $\iff$ $f^{-1}(A)$ is open in $M$. Now $f,f^{-1}$ are inverse homeomorphisms with this new topological structure.
Next, if $(U_\alpha,\phi_\alpha)$ is a cover of $M$ by open sets $U$ and homeomorphisms $\phi_\alpha:U_\alpha\to \tilde U\subset \mathbb R^n$ gives the manifold structure on $M$, then define $(V_\alpha,\psi_\alpha)$ as $V_\alpha = f(U_\alpha)$ and define $\psi_\alpha$ to be $\phi_\alpha\circ f^{-1}$ . The $\psi_\alpha$ are homeomorphisms.
Now I will rest and ask you to work. Prove four things:

In the second paragraph, prove that $f$ and $f^{-1}$ are homeomorphisms with the topology I just defined on $A$.
In the third paragraph, prove that the $(V_\alpha,\psi_\alpha)$ are actually valid charts giving $A$ a manifold structure
Prove that $f$ gives a diffeomorphism with this manifold structure on $A$
Prove that this is the unique manifold structure that gives a diffeomorphism (First prove that the topology cannot be any different and next prove that the manifold structure cannot either.

Let me know if you have trouble somewhere or need additional help.


TITLE: Maximize multivariable integer function
QUESTION [0 upvotes]: Let $L\ge 2$ be an integer, $a_0=10$, $a_L=2$, and $a_1,...,a_{L-1}\ge 2$ be integers summing to $36$. Maximize $\sum_{i=0}^{L-1}a_ia_{i+1}$ over $L$ and $a_1,...,a_{L-1}$.
I think the maximum is $556$ achieved at $L=3$ and $(a_1,a_2)=(22,14)$. It’s easy to show if $L\le 3$. For $L>3$, I tried to increase the sum by using $a_1,...,a_{L-3},a_{L-2}+a_{L-1}$ instead of $a_1,...,a_{L-1}$, but it doesn’t work for arbitrary $a_1,...,a_{L-1}$, so I tried to first increase the sum by swapping $a_i$ with $a_j$ when they were out of order in some sense, like the proof of the rearrangement inequality, but I couldn’t figure out how. Maybe there is a way using calculus.

REPLY [0 votes]: For a particular numerical value of $L$, this can be formulated and solved as a Mixed Integer non-convex Quadratic Programming problem, a.k.a.non-convex MIQP, using readily available non-convex MIQP solvers.
CPLEX, using optimalitytarget = 3 can be used to (try to) solve this to global optimality., or using  optimalitytarget = 2 to local optimality. BARON could also be used to solve for the global optimum.  Whether or not the global optimum is found and claimed to be found depends on available time and memory. It could be fast or take a very long time.


TITLE: Parameterize a circle (given x, y, z cooridnates on its circumference) in 3D
QUESTION [0 upvotes]: I have a circle defined by a set of $x, y$ and $z$ coordinates. The circle exists across $3$ planes. I would know how to parameterize a $2$-d circle (say in just the $x$-$y$ plane) into polar coordinates, but not in $3$-d. How do I do this? I visualized the set of $x, y$ and $z$ coordinates in Paraview so I could see the orientation of the circle; this can be seen below.
I need to paramterize this circle because I need to define the $x, y$ and $z$
 points I was given as angles around the circumference of that circle.

(Note the circle of interest is the translucent white object in the centre of the cube domain)

REPLY [1 votes]: Find from your data the center $O$ of the circle and then two points $A$, $B$ on the circle such that $OA\perp OB$. The parametric equation of the circle is then:
$$
(x,y,z)=O+(A-O)\cos\phi+(B-O)\sin\phi.
$$
Inserting here the coordinates of a given point of the circle you can solve for $\cos\phi$ and $\sin\phi$. Of course point $A$ corresponds to $\phi=0$.


TITLE: From a group of three biologists, two physicists and one mathematician, a committee of two people is to be randomly selected.
QUESTION [1 upvotes]: From a group of three biologists, two physicists and one mathematician, a committee of two people is to be randomly selected. Denote by $X$ the random variable representing the number of biologists and by $Y$ the random variable representing the number of physicists on the committee. Calculate: $f_{X, Y}$, $f_{X}$, $f_{Y}$.
Attempt
$$f_{X}=\frac{\binom{3}{1} \binom{2}{1}}{\binom{5}{2}}+\frac{\binom{3}{1} \cdot 1}{\binom{5}{2}}$$
$$f_{Y}=\frac{\binom{2}{1} \binom{3}{1}}{\binom{5}{2}}+\frac{\binom{2}{1} \cdot 1}{\binom{5}{2}}$$
Then, $$f_{X, Y}=f_{X}+f_{Y}$$
Am I understanding the problem?

REPLY [0 votes]: No that is not correct. You are supposed to find probability mass function $ \small f_X(x), f_Y(y), f_{XY} (x, y)$. Below is a tabular representation of what you need to come up with,
$ \small \begin{array}{|c|c|c|}
  \hline
   & Y=0 & Y=1 & Y=2 & f_X\\
 \hline
  X=0 &  &  \\
  \hline
  X=1 &  &  \\
  \hline
X=2 &  &  \\
  \hline
f_Y &  & & & - \\
  \hline
\end{array}$
We have $3$ biologists, $2$ physicists and $1$ mathematician. We can first find pmf $ \small f_{XY}(x, y)$. For example,
$\small f_{XY}(X = 0, Y = 0) = 0 ~ $ as a two member committee cannot be formed with just mathematician.
$ \displaystyle \small f_{XY}(X = 1, Y = 0) = {3 \choose 1} {1 \choose 1} / {6 \choose 2} ~ $ as we choose one of the biologists and the other member is mathematician.
$ \displaystyle \small f_{XY}(X = 2, Y = 0) = {3 \choose 2}  / {6 \choose 2}$
Similarly find
$\small f_{XY}(X = 0, Y = 1), f_{XY}(X = 1, Y = 1), f_{XY}(X = 0, Y = 2)$
Then,
$ \small f_Y(Y = 0) = f_{XY}(X = 0, Y = 0) + f_{XY}(X = 1, Y = 0) + f_{XY}(X = 2, Y = 0)$
You can similarly find $ \small f_Y(Y = 1), f_Y(Y = 2), f_X(X = 0), f_X(X = 1)$ and $ \small f_X(X = 2)$


TITLE: Stars and Bars Equivalence
QUESTION [2 upvotes]: My textbook says that "statement" executes ${10+(3-1)}\choose{3}$ times for the following pseudocode without giving any details on why (I know that $n=10$ and $r=3$). I have verified this to be correct with a simple script.
for i = 1 up to 10
    for j = 1 up to i
        for k = 1 up to j
            statement

How can I visualize how many times "statement" is run through the use of stars and bars or any other combinatorial argument?
This is not the same as this or this - they are only calculating how many times it is run without using stars and bars.

Stars and Bars:

If there are $n$ identical objects and $r$ distinct containers, then there are ${n + (r-1)}\choose{r}$ possibilities to distribute the objects.
If there are $5$ objects and $3$ containers, then $3$ objects in container 1, $1$ object in container 2 and $1$ object in container 3 can be represented with stars and bars by $***|*|*$

REPLY [2 votes]: Let $10 =a+b+c+d+1,\ i=b+c+d+1,\ j=c+d+1,\ k=d+1$.
where $a,b,c,d$ each represents the 'stars' in every container:

For example in $(∗∗∗|∗|∗∗|∗∗∗)$ $a=3, b=1, c=2, d=3$

Then $\mathrm{the\ number\ of\ permutations} = {9+(4-1)\choose (4-1)} = {12\choose 3} $which apparently is the same as the given one in the textbook.
Then $i$ will be any number from $1$ to $10$, $j$ will be any number from $1$ to $i$, and $k$ will be any number from $1$ to $j$.

*Your formula for stars and bars is not correct.

Source: Brilliant.org
The number of ways to place $n$ indistinguishable balls into $r$ labelled urns is
$$ \binom{n+r-1}{n} = \binom{n+r-1}{r-1}. \ _\square $$

which is not the same as the formula you have given, i.e. ${n + (r-1)}\choose{r}$