TITLE: Finding all $z \in \Bbb C$, expressed in the form $z = a + ib$, with $a, b \in \Bbb R$, satisfying equation $z^3 = -i$
QUESTION [4 upvotes]: I'm trying to find all $z \in \Bbb C$, expressed in the form $z = a + ib$, with $a, b \in \Bbb R$, satisfying equation $z^3 = -i$.
I've figured out that
$$(a + ib)^3 = -i \Longleftrightarrow a + ib = \sqrt[3]{-i} \Longleftrightarrow a + ib = -i$$
and also that
$$(a+ib)^3 = a^3 + 3a^2bi + 3ab^2i^2 +i^3b^3$$
but I'm not sure how to proceed from here.
I think I'm supposed to use synthetic division to find the other roots, but not sure what factor to use. I tried $a + ib + i$ and that didn't work.
Any ideas?

REPLY [2 votes]: Without using polar representation:
Note that
$i^3 = i^2 i = (-1)i = -i; \tag 1$
thus $i$ is a solution of
$z^3 = -i, \tag 2$
which may be re-written as
$z^3 + i = 0; \tag 3$
we may seek the other zeroes of (2)-(3) via synthetic division; we seek
$z - i \overline{)z^3 + i}; \tag 4$
we have:
$z \; \text{into} \; z^3 \; \text{yields} \; z^2; \tag 5$
$z^2 \times (z - i) = z^3 - iz^2; \tag 6$
$z^3 + i - (z^3 - iz^2) = iz^2 +  i = i(z^2 + 1) = i(z + i)(z - i) ; \tag 7$
now we can skip to the chase by noticing that
$(z - i) \overline{)i(z + i)(z - i)} = i(z + i) = iz - 1; \tag 8$
therefore the quotient should be
$(z - i) \overline{)z^3 + i} = z^2 + iz - 1; \tag 9$
we check:
$(z - i)(z^2 + iz - 1) = z^3 + iz^2 - z - iz^2 + z + i = z^3 + i; \tag{10}$
the quotient (4) is thus the quadratic $z^2 + iz - 1$; we use the quadratic formula:
$z = \dfrac{1}{2} (-i \pm \sqrt{i^2 - 4(1)(-1)}) = \dfrac{1}{2}(-i \pm \sqrt 3), \tag{11}$
which are easily checked to satisfy (3); I leave that simple task to my readers.  As well as the tasks of converting our roots to totally proper $a + bi$ form.


TITLE: For a sequence of positive real numbers converging to a limit (which is not equal to 0), show that infimum > 0
QUESTION [0 upvotes]: I think to do this I either need to prove the infimum is a minumum or a limit but I'm not sure how.
I have tried this:
By definition of the infimum:
" Let A be a subset of the real numbers and b be a real number.
Then b is the infimum of A if:
(i) b≤a for all a ∈ A
(ii) c≤a for all a ∈ A will imply that c≤b."
We know for all a ∈ A, a>0.
If we consider the second case, this means the infimum (b) is such that 0≤b.
We can assume for a contradiction that b = 0 in order to leave just the case 0 < b.
b cannot be the minimum of the set, as all a > 0.
because all the terms are positive, for the infimum to be 0, it means the limit of the series is zero (and the sequence must be decreasing), but L cannot be equal to 0 so we have a contradiction.
This highlighted bit is the jump I'm struggling with.

REPLY [0 votes]: When studying new mathematical material, an attempt must be made be to write down 'the setup' in a mathematically precise way, and to avoid using logical arguments that use the informal language of, say, English.
The setup:
$\tag 1 \lim\limits_{n \to +∞} a_n = L, \text{ with } L \gt 0 \text{ and } a_n \gt 0$
$\tag 2 A = \{ a_n \, | \, n \in \mathbb N\}$
$\tag 3 b = \text{inf}(A)$
Assume $b = 0$.
Using open intervals of the form $(0, \varepsilon)$, a strictly decreasing sequence  $b_n$ with $b_n \in A$ can be constructed such that
$\tag 4 \lim\limits_{n \to +∞} b_n = 0$
Now $b_n$ might not be a subsequence of $a_n$, but you can take another pass over it, throwing out terms if necessary to get the desired subsequence. Since every subsequence of $(a_n)_{ \, n \in \mathbb N} $must have the same limit, $L = 0$, we arrive at a contradiction.
Note that you can construct $b_n$ as a subsequence on the first pass, but thought it might be easier to follow by breaking it into two steps. The basic truth is that for any finite set of positive numbers, an $\varepsilon \gt 0$ can be found such that the open interval $(0, \varepsilon)$ is disjoint from that set.

The OP states in bold

it means the limit of the series is zero (and the sequence must be
  decreasing)

Let $(a_n)_{ \, n \in \mathbb N} $ be defined by
$$
    a_n  = \left\{\begin{array}{lr}
        \frac{1}{n+1}, & \text{for } n \text{ even }\\
       \frac{1}{n-.5}, & \text{for } n \text{ odd }
        \end{array}\right\}
$$
This sequence of positive numbers converges to $0$, but it is not a decreasing sequence
(see the wikipedia definition).


TITLE: Rotating Frame Transformation for three level system
QUESTION [2 upvotes]: I am trying to solve a task from an exercise sheet:
We consider a three-level system ($\Lambda$-configuration) with three eigenstates $|1\rangle$, $|2\rangle$ and
$|3\rangle$. The levels $|1\rangle$ and $|3\rangle$ are coupled by a probe beam with
Rabi frequency $\Omega_d$ and angular frequency $\omega_d$. Additionally, Level $|3\rangle$ is coupled to $|2\rangle$ by
a strong drive field with Rabi frequency $\Omega_p$ and angular frequency $\omega_d$. We denote the
decay rates for spontaneous emission between the levels by
$\gamma_{31}$ and
$\gamma_{32}$.
a) Construct the Hamiltonian of the system and show that in an appropriate rotating
frame it can be written as
$$ H = \left(
\begin{matrix}
    0 & 0 & \Omega_p \\
    0 & \Delta_p-\Delta_d & \Omega_d \\
    \Omega_p & \Omega_d & \Delta_p \\
    \end{matrix}
\right) $$
where $\Delta_p$ = $\omega_3-\omega_1-\omega_p$ and $\Delta_d = \omega_3-\omega_2-\omega_d$. It is useful to introduce the operators
$\sigma_{13} = |1\rangle\langle3| $ and $\sigma_{23} = |2\rangle\langle3|$.

I started by setting up the free and the interaction Hamiltonian as:
$ H = H_0+H_{int}$
and
$H_0 = \hbar(\omega_1 \sigma_{11} +\omega_2 \sigma_{22} +\omega_3 \sigma_{33})$
With the rotating wave approximation, I can ommit the quickly oscillating terms in the interaction Hamiltonian:
$H_{int} = \hbar\left(\Omega_p\left( \sigma_{13}e^{i\omega_p t} + \sigma_{31}e^{-i\omega_p t} \right)+ \Omega_d\left( \sigma_{23}e^{i\omega_d t} + \sigma_{32}e^{-i\omega_d t} \right) \right)$
Now putting this into a Matrix and setting $\omega_1 = 0$ I get:
$$ H = \left(
\begin{matrix}
    0 & 0 & \Omega_pe^{i\omega_p t} \\
    0 & \omega_2 & \Omega_de^{i\omega_d t} \\
    \Omega_pe^{-i\omega_p t} & \Omega_de^{-i\omega_d t} & \omega_3 \\
    \end{matrix}
\right) $$
But is this what I am supposed to do here? I feel like I am missing something because first I am not using the spontaneous emission rates in the Hamiltonian at all and second I am left with those exponential functions in the off-diagonal terms and the terms on the diagonal are not in the required form.
My guess would be to find some rotation Matrix such that $H' = RHR^\dagger$ but I have no idea how this matrix would look like.
I would really appreciate some help on what I'm missing here, google and my script didn't really help me out this time...

REPLY [0 votes]: Transforming to a rotating frame means that you use a time-dependent unitary transformation $U(t)$. The transformed Hamiltonian is
$$ H'(t) = U(t) H(t) U(t)^\dagger + \mathrm i \hbar\, \dot U(t) U(t)^\dagger $$
(see for example this answer for an explanation of the second term).
In your concrete problem, since a) it is a homework exercise and therefore hopefully not too complicated and b) we really only want to get rid of some phases in the Hamiltonian, a good ansatz for the transformation would be
$$ U(t) = \operatorname{diag}\bigl( \mathrm e^{\mathrm i \alpha t}, \mathrm e^{\mathrm i \beta t}, \mathrm e^{\mathrm i \gamma t} \bigr) . $$


TITLE: Light-cone coordinates to quantize strings?
QUESTION [1 upvotes]: Zweibach, in A First Course in String Theory said: "We now discuss a coordinate system that will be extremely useful in our study of string theory, the light-cone coordinate system", then he mentioned that this coordinate system will be used to quantize strings.
My question: What's so special about light-cone coordinates/gauge, so they have used them to quantize strings? Why are they "extremely useful"?

REPLY [2 votes]: Light-cone (LC) coordinates appear in bosonic string theory in two places:

In the World-sheet (WS). To quantize a physical system, one should always start by analyzing the corresponding classical system. The classical equation of motion of the free bosonic string is the wave equation $$\Box X ~\approx~ 0 $$ in 1+1D. It is a linear 2nd-order PDE, whose full/general solution $$X ~\approx~  X_L(\sigma^+)+X _R(\sigma^-)$$  is a sum of arbitrary left- and right-moving solutions, each of which only depends on one of the two WS LC coordinates $\sigma^{\pm}$. The WS LC coordinates are the characteristics of the PDE. This is essentially the main reason for the prominent role played by the WS LC coordinates $\sigma^{\pm}$.

In the target space (TS). The bosonic string action has WS reparametrization gauge symmetry. One can use modern Lorentz-covariant BRST quantization, but it is easier to work in a particular gauge. Any admissible TS gauge choice will in principle do (in the sense that gauge-invariant physical observables do not depend on gauge-choice), but the TS LC gauge
$$X^+(\tau,\sigma)~=~f(p^+(\tau))\tau\qquad\text{and}\qquad P^+(\tau,\sigma)~=~p^+(\tau),$$
is a convenient choice to effectively separate physical and unphysical degrees of freedom (DOF) and minimize the needed algebra, cf. e.g. this Phys.SE post.


TITLE: Why are random variables measurable?
QUESTION [8 upvotes]: Why do we require random variables to be measurable?
The only reason I can think of is being able to identify which event happened knowing that the value of random variable (and therefore we can also calculate the probability of random variable assuming that particular value)?
This is what the definition of random variable basically says where it requires the random variable function to be measurable. Do the reasons I gave above make sense?

REPLY [11 votes]: The reason you give in parentheses is the correct one - we need to be able to compute probabilities with random variables, and the only way to do this is to find the pre-image of sets and compute probabilities with them.  This requires that the pre-image of (measurable) sets be measurable, which is the definition of a measurable function.
To be more precise, suppose $(\Omega_1,\mathcal{F}_1,P)$ is your probability space, $(\Omega_2,\mathcal{F}_2)$ is your "target" space (this is usually taken to be $\mathbb{R}$ with its Borel sets).  Then, suppose we want $X:\Omega_1\rightarrow \Omega_2$ to be a random variable.  This means that we want to be able to measure the probability of events like $\{X\in A\}$ where $A\in \mathcal{F}_2$.  Well, the way we do this is to find $X^{-1}(A)$, and try to compute $P(X^{-1}(A))$.  We can only do this if $X^{-1}(A)\in \mathcal{F}_1$, in other words, if $X$ is a measurable function.


text:
\DOC STRIP_GOAL_THEN

\TYPE {STRIP_GOAL_THEN : thm_tactic -> tactic}

\SYNOPSIS
Splits a goal by eliminating one outermost connective, applying the
given theorem-tactic to the antecedents of implications.

\KEYWORDS
theorem-tactic.

\DESCRIBE
Given a theorem-tactic {ttac} and a goal {(A,t)}, {STRIP_GOAL_THEN} removes one
outermost occurrence of one of the connectives {!}, {==>}, {~} or {/\} from the
conclusion of the goal {t}.  If {t} is a universally quantified term, then
{STRIP_GOAL_THEN} strips off the quantifier:
{
      A ?- !x.u
   ==============  STRIP_GOAL_THEN ttac
     A ?- u[x'/x]
}
\noindent where {x'} is a primed variant that does not appear free in the
assumptions {A}.  If {t} is a conjunction, then {STRIP_GOAL_THEN} simply splits
the conjunction into two subgoals:
{
      A ?- v /\ w
   =================  STRIP_GOAL_THEN ttac
    A ?- v   A ?- w
}
\noindent If {t} is an implication {"u ==> v"} and if:
{
      A ?- v
  ===============  ttac (u |- u)
     A' ?- v'
}
\noindent then:
{
      A ?- u ==> v
  ====================  STRIP_GOAL_THEN ttac
        A' ?- v'
}
\noindent Finally, a negation {~t} is treated as the implication {t ==> F}.

\FAILURE
{STRIP_GOAL_THEN ttac (A,t)} fails if {t} is not a universally quantified term,
an implication, a negation or a conjunction.  Failure also occurs if the
application of {ttac} fails, after stripping the goal.

\EXAMPLE
When solving the goal
{
  # g `n = 1 ==> n * n = n`;;
  Warning: Free variables in goal: n
  val it : goalstack = 1 subgoal (1 total)

  `n = 1 ==> n * n = n`
}
\noindent a possible initial step is to apply
{
  # e(STRIP_GOAL_THEN SUBST1_TAC);;
  val it : goalstack = 1 subgoal (1 total)

  `1 * 1 = 1`
}
\noindent which is immediate by {ARITH_TAC}, for example.

\USES
{STRIP_GOAL_THEN} is used when manipulating intermediate results (obtained by
stripping outer connectives from a goal) directly, rather than as assumptions.

\SEEALSO
CONJ_TAC, DISCH_THEN, GEN_TAC, STRIP_ASSUME_TAC, STRIP_TAC.

\ENDDOC

subset_name: curated


TITLE: How should you explain parallel transport to undergraduates?
QUESTION [33 upvotes]: The title is a bit deceiving, because what I really mean is the parallel transport that corresponds to the Levi–Civita connection.
This is in the vein of many other questions on mathoverflow:

What is the Levi-Civita connection trying to describe?

What is torsion in differential geometry intuitively?

Rolling without slipping interpretation of torsion

But the focus is different.
Let me summarize my understanding given answers in the questions above:
(Very Rough) Summary of Answers in Previous Questions

There exists an interpretation in terms of $G$-structures, as in the chosen answer (by Chris Schommer-Pries) to What is torsion in differential geometry intuitively?.
There exists an interpretation as having some universal property, as described in the top answer (by Robert Bryant) to What is the Levi-Civita connection trying to describe?.
There exists a deceiving but appealing interpretation by parallelograms whose sides don't really lie in the same space, as in the answer by Gabe K to What is the Levi-Civita connection trying to describe?. (Gabe K did a heroic effort to make sense of the nonsensical diagram, and I thank him dearly.)
There exists an interpretation regarding rolling the shape on a surface (Rolling without slipping interpretation of torsion).

But ultimately, none of that is something that I can intuitively sell to an undergraduate, and by undergraduate I really mean my heart. In the bottom of my heart, I need a better explanation, one that starts with desirable properties, and then proceeds through existence and uniqueness.
Outline of the Type of Intuition I Desire
I want to start with some desirable behaviors, which I allow to be external (i.e, to reference a given embedding of the Riemannian manifold into $\mathbb{R}^n$), and then say that the only notion of parallel transport that satisfies these conditions must be the Levi–Civita connection. (Any reasonable notion of parallel transport will respect the metric, so I'm really thinking of the torsion-free condition.)
A base case of a desirable condition is that for the Riemannian manifold $\mathbb{R}^n$, parallel transport is the trivial thing. (If one identifies the tangent bundle with $\mathbb{R}^n\times\mathbb{R}^n$ then for any path $\gamma$ the parallel transport of the tangent vector $(\gamma(0),v)$ at $\gamma(0)$ to $\gamma(1)$ via $\gamma$ is the tangent vector $(\gamma(1),v)$ at $\gamma(1)$.)
Next, we would like some way to generalize to a general Riemannian manifold. Let $(M,g)$ be a Riemannian manifold, and let $p\in M$ be a point. Then by the implicit function theorem we can have a chart $f:V\rightarrow U\subset \mathbb{R}^d$ where $0\in V\subset \mathbb{R}^n$, and $p\in U\subset M$, such that $f(0)=p$ and such that $f$ is the identity on the first $n$ coordinates.
My next thought is to look at the most intuitive case of torsion-freeness, which is the case of commuting fields $X$ and $Y$. By change of coordinates, we can assume WLOG that on $V$ the vector fields $X$ and $Y$ are defined via the constant functions $X(v)=e_1$ and $Y(v)=e_2$. One can then express $X$ on $Y$ on $M$ via the derivative of $f$.
But I'm missing multiple components to proceed.
So let me ask this in terms of several more explicit questions.
Questions

If a connection satisfies that $\nabla_XY=\nabla_YX$ for any commuting set of vector fields $X$ and $Y$, then is it torsion-free? (In other words, if you're torsion-free on commuting vector fields, are you torsion-free for all vector fields?)
What intuitive desirable condition (that is allowed to use a given embedding of $M$ into some $\mathbb{R}^d$), combined with, or perhaps generalizing the desired behavior of parallel transport on Euclidean space, would uniquely determine it as satisfying $\nabla_XY=\nabla_YX$ for commuting vector fields? (Perhaps something about geodesics? Or volumes? I don't really know what's the missing component here.)
I feel like Ben McKay's answer to What is the Levi-Civita connection trying to describe? is coming close to what I want, but I did not get to the bottom of it. It appeared at first that he was saying that the Levi–Civita parallel transport is simply parallel transporting in the ambient space, and then projecting to the tangent plane. But in retrospect, my interpretation is clearly wrong. (Imagine for example an upward pointing vector on the equator of a sphere, being parallel transported to the top. If you parallel transport in $\mathbb{R}^3$ you'll get a vector pointing up, which projected to the tangent space will be the $0$ vector.)
A little more vaguely, in case you have an entirely different notion in mind, how would you explain parallel transport to the undergraduate in your heart?

REPLY [0 votes]: If intuition is what you want then I think the essence of the matter is best expressed using locally flat coordinates at each point as follows. At any point $p$ in a Riemannian (or pseudo-Riemannian, for those of us who use this in the context of general relativity) manifold, one can always choose "locally flat coordinates" (lfc) such that the metric components gave vanishing first partial derivatives at $p$. The (Levi-Civita) covariant derivative of a vector (or tensor) at $p$ is simply the partial derivative with respect to any lfc. A vector is parallel transported along a curve if at each point in lfc its components have vanishing derivative with respect to the curve parameter. This is of course not a practical definition, since the lfc are different at each point. For practicality, one can use the connection components in any coordinate system. Note however that if the curve is a geodesic, then there exist coordinates that cover a neighborhood of the curve and are locally flat at every point on the curve. In that case, parallel transport just keeps all the components of the vector constant in such a coordinate system. An example is transport around the equator of the 2-sphere using standard spherical polar coordinates.