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Questions tagged [mathematical-physics]

DO NOT USE THIS TAG just because your question involves math! If your question is on simplification of a mathematical expression, please ask it at math.stackexchange.com. Mathematical physics is the mathematically rigorous study of the foundations of physics, and the application of advanced mathematical methods to problems in physics. Examples include partial differential equations (PDEs), variational calculus, functional analysis, and potential theory.

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7
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2answers
645 views

What type of non-differentiable continuous paths contribute for the path integral in quantum mechanics

Consider the path integral for a 1D particle subjected to a potential $V(x)$ in imaginary time $$ \int_{x(0)=x_0}^{x(T)=x_T} [dx] \, e^{- \int_0^T d\tau \left[\frac{1}{2}\dot{x}^2 + V(x(\tau))\right]}...
5
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1answer
430 views

Rigorous version of field Lagrangian

In Classical Mechanics the configuration of a system can be characterized by some point $s\in \mathbb{R}^n$ for some $n$. In particular, if it's a system of $k$ particles then $n = 3k$ and if there ...
1
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1answer
66 views

Is the string-net model Hermitian?

In Kitaev and Kong's paper, they define the Hermitian inner product on morphism spaces in Eq. (11). My question is that: Given that F symbols satisfy the pentagon identity, does that the string-net ...
1
vote
0answers
32 views

Spectral representation of a BCS gap function

I am playing with the spectral representation of a BCS gap function and I have trouble verifying causality properties. I find a divergence and I don't know what is the problem. Assume the gap ...
5
votes
1answer
432 views

General derivative of the exponential operator w.r.t. a parameter

I am interested in the calculation of the general $N$th derivative w.r.t. a parameter $\lambda$ of a quantum mechanical exponential operator with the following structure: \begin{equation*} \frac{\...
11
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1answer
157 views

Metric interpretation of self-adjoint extensions?

I am wondering if beyond physical interpretation, the one dimensional contact interactions (self-adjoint extensions of the the free Hamiltonian when defined everywhere except at the origin) have a ...
41
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8answers
5k views

Negative probabilities in quantum physics

Negative probabilities are naturally found in the Wigner function (both the original and its discrete variants), the Klein paradox (where it is an artifact of using a one-particle theory) and the ...
2
votes
1answer
303 views

Equivalent system in Centre manifold theory

I was studying the centre manifold theory. It says (see Kuznetsov (pdf) page 155, theorem 5.2) that the system on the left side of the picture is topologically equivalent to the one on the right. $ \...
13
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4answers
8k views

A book on quantum mechanics supported by the high-level mathematics

I'm interested in quantum mechanics book that uses high level mathematics (not only the usual functional analysis and the theory of generalised functions but the theory of pseudodifferential operators ...
2
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1answer
116 views

Gauge-invariance of Lagrangians

I am rereading David Bleecker's Gauge Theory and Variational Principles, and I have realized I don't understand something. The offending part is in 3.3 (page 50-52), however I am reproducing the ...
0
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1answer
34 views

Is the disposition of $1$s and $0$s when writing orthogonal ket vectors purely conventional?

If I want to define the basis in the form of $4$-vectors, how do I proceed to make sure they are orthonormal with one $1$ and three $0$ in each vector? Is it just by convention? Does it matter if I ...
1
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0answers
58 views

Looking for lecture videos that follow Arnold's Mathematical Methods of Classical Mechanics

I'm an undergrad and I'm looking for lecture videos (on youtube and such) that follow this textbook. My course roughly follows it, but glosses over some mathematical details that I feel would be ...
0
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1answer
71 views

Physical meaning of theorem

This is the image of theorem from V.I Arnold's Mathematical method of mechanics. I understood the example given in text. But I want to know what is physical meaning of example? Can anybody help?
6
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1answer
92 views

Distinct choice of partition in the Path Integral

Practically all books in Quantum Mechanics and Quantum Field Theory define the non-relativistic path integral by taking one interval $[a,b]$ and breaking it up into $N$ subintervals of equal length. ...
29
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4answers
12k views

Book covering differential geometry and topology for physics

I'm interested in learning how to use geometry and topology in physics. Could anyone recommend a book that covers these topics, preferably with some proofs, physical applications, and emphasis on ...
2
votes
1answer
137 views

Why there's a Lorentz inner product in the unitary representations of the translation group?

Consider Minkowski spacetime. Its translation group is just the additive group $\mathbb{R}^4$. This is an abelian locally compact group. Next, consider one unitary representation $T : \mathbb{R}^4\to ...
5
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1answer
254 views

Set theories used in mathematical physical theories

For most theorems, this is unimportant - a typical problem will have the same solutions in NFU, ZFC or some other theory, since said theories got popular precisely because they give pretty much the ...
11
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6answers
4k views

Mathematical rigorous introduction to solid state physics

I am looking for a good mathematical rigorous introduction to solid state physics. The style and level for this solid state physics book should be comparable to Abraham Marsdens Foundations of ...
0
votes
1answer
35 views

Meaning and Origin of an Expression which Involves Virtual Displacement

As an additional point of confusion related to the answer given here: Confusion with Virtual Displacement I have encountered the following expression in my study of virtual displacements. $$\delta{...
2
votes
1answer
130 views

Taking a trace using a continuous spectrum of eigenstates

This may be a simple question, but I have not been able to find an adequate discussion in any source that quite answers it. In many cases in quantum mechanics, traces are evaluated using the ...
3
votes
3answers
2k views

Virtual displacement and generalized coordinates

I have a doubt regarding the expression of a virtual displacement using generalized coordinates. I will state the definitions I'm taking and the problem. The system is composed by $n$ points with ...
2
votes
2answers
235 views

Confusion with Virtual Displacement

I have just been introduced to the notion of virtual displacement and I am quite confused. My professor simply defined a virtual displacement as an infinitesimal displacement that occurs ...
2
votes
1answer
126 views

How to make sense of $\mathcal{I}^-$ as a Cauchy surface rigorously?

In some references, like Hawking's derivation of black hole radiation, one considers that $\mathcal{I}^-$ is a Cauchy surface. One recent reference with such a claim is the paper "Soft Hair on Black ...
1
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0answers
109 views

Pure math courses for physicists: Topology [closed]

I'm in my bachelor in physics. In a couple of weeks I start my last year, and I'm interested in taking some pure math courses. As you see, I like the theoretical point of view, but I don't know if the ...
-1
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1answer
49 views

Negative unity matrix not hermitian? (stabilizer formalism)

I read the section in the attached picture about the stabilizer formalism and was wondering about the last sentence in the pic. It says that all operators of the stabilizer group are hermitian, ...
1
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0answers
26 views

conventional matrix notation for distance interval

Why matrix notation for distance interval is represented by this? $$g_{\mu \nu}\Delta X^{\mu}\Delta X^{\nu}=(\Delta X)^Tg (\Delta X)=\Delta X^{\mu}\Delta X^{\nu}g_{\mu \nu}$$ Could you explain ...
-1
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1answer
131 views

What is the meaning of “representation of the canonical commutation relation in the form of Heisenberg for symplectic locally convex space”?

What is the meaning of "representation of the canonical commutation relation in the form of Heisenberg for symplectic locally convex space"?
2
votes
1answer
87 views

Post-measurement density matrix derivation

This is something standard, by I'm trying to redo this with spectral theory. Suppose we start with the usual postulates of quantum mechanics: States are unit rays on a separable Hilbert space. In ...
0
votes
1answer
52 views

Position of a particle sliding down an arbitrary curve as a function of time

Given a curve in a frictionless environment with parameterization $\displaystyle \mathbf{r}(\theta)=x(\theta)\hat{\mathbf{i}}+y(\theta)\hat{\mathbf{j}}$ for $\theta\in[0,\theta_f]$, how can I find the ...
2
votes
1answer
117 views

Utility of the time-ordered exponential

Is the time-ordered exponential $$ \mathcal{T}\exp\left\{-i\int_{t_0}^tdt'V(t')\right\}\tag{1} $$ just a mnemonic device for the series $$ \begin{aligned} 1 + (-i)\int_{t_0}^tdt_1 \, V(t_1) +{} &...
11
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2answers
356 views

Does mathematical sloppiness in quantum mechanics ever produce incorrect predictions?

Does mathematical sloppiness in standard quantum mechanics ever produce predictions that don't pan out? I'm not talking about things like the WKB approximation, but instead subtle functional analytic ...
2
votes
1answer
99 views

Trace over configuration basis

Let us take a many-body quantum system, whose phases in the configuration basis are labeled by $\mathbf {\hat q}=(q_1,\cdots, q_N)$ and momenta $\mathbf {\hat p}=\left(-i\frac{\partial}{\partial \hat ...
2
votes
0answers
55 views

Pictures of Different Coordinate Systems in General Relativity

In General Relativity by Woodhouse there are the three following diagrams in Chapter 9 about Black Holes. Despite a (very brief) description of these diagrams in the book itself, I am struggling to ...
10
votes
3answers
2k views

Does Hestenes Zitterbewegung Explain why complex numbers appear in QM?

This question may fit better in the discussion of "Why Complex variables are required by QM?", but it also relates to the extent to which arguments by Hestenes are accepted in mainstream physics and ...
18
votes
2answers
2k views

How does one measure space-like geodesics? Or: What is the physical interpretation of space-like geodesics?

In general relativity, time-like geodesics are the trajectories of free-falling test particles, parametrized by proper time. Thus, they are easy to interpret in physical terms and are easy to measure (...
2
votes
1answer
131 views

Classical Green's function

I am confused about a classical calculation involving Green's functions. Suppose $\frac{d^2}{dx^2}G+k^2G=\delta(x)$. Then Fourier theory leads, apart from numerical factors, to $$G(x)=\int \frac{e^{...
6
votes
3answers
151 views

Domains of $H$ and $U(t) = \exp(-iH t )$

I am not so familiar with functional analysis. But in my impression, the Hamiltonian $H$ is often not defined everywhere on the Hilbert space. On the other hand, the time evolution operator $U(t)$, ...
7
votes
1answer
112 views

2D global conformal transformations and the $z= \frac1w$ argument

For instance in Blumenhagen's CFT, there is a standard argument which determines that globally defined conformal transformations on the Riemann sphere where $$l_n = -z^{n+1} \partial_z$$ is an ...
3
votes
1answer
83 views

Auxiliary Grassmann variables in supergeometry

I was reading on super geometry and how it is used to model fermions and supersymmetry in classical field theory. In texts like [1] or [2] the authors introduced auxiliary Grassmann odd variables to ...
1
vote
1answer
87 views

Sound wave equation: Neumann boundary conditions

In this paper it's described the solution of the damped wave equation in cylindrical coordinates $$ \nabla^2\left(c^2\rho_1+\nu\frac{\partial\rho_1}{\partial t}\right)-\frac{\partial^2\rho_1}{\...
2
votes
0answers
66 views

Motivating the Unintuitive Properties of Spinors

In the usual treatment of (Dirac) spinors, one usually starts with "factoring" the energy-momentum relation, deducing the properties of the $\gamma$ matrices by requiring the cross terms to cancel, ...
1
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0answers
47 views

self-adjoint extension of the momentum operator in an infinitely deep potential

Theta parameter arises when calculating self adjoint extensions of the momentum operator of a particle in an infinnitely deep potential, what does this means physically?
6
votes
2answers
302 views

Is this actually the rigorous definition of the path integral in Quantum Mechanics?

Let a quantum system with a single degree of freedom be given. We want to define the path integral so that we get the representation for the propagator as $$\langle q' |e^{-iHT}|q\rangle=\int_{x(a)=...
1
vote
0answers
34 views

What is the magic behind Sector Decomposition?

I have a question regarding Sector Decomposition, which is briefly introduced in this paper arXiv: 0803.4177. So far I played around with a toy example and applied the Sector Decomposition method to ...
2
votes
1answer
78 views

Joint Spectral Measure theorem

I want to gain an intuition to understand the joint spectral measure theorem. In the case that operators involved in this theorem have purely discrete spectrum, the theorem should be reduced to the ...
3
votes
1answer
205 views

Domain of symmetric momentum operator vs self-adjoint momentum operator

Is there an example of a function that is not in the domain of the 'naive' symmetric (but not self-adjoint) momentum operator $p:=-i\frac{d}{dx}$ but is in the 'true' self-adjoint momentum operator $p:...
40
votes
4answers
3k views

Hilbert space of harmonic oscillator: Countable vs uncountable?

Hm, this just occurred to me while answering another question: If I write the Hamiltonian for a harmonic oscillator as $$H = \frac{p^2}{2m} + \frac{1}{2} m \omega^2 x^2$$ then wouldn't one set of ...
3
votes
1answer
109 views

Question about characteristics and classification of second-order PDEs

I know this question is quite maths-focused; however, it relates closely to numerical methods that are used to solve Physics problems (for example in Fluid Dynamics/CFD). I asked the same question ...
2
votes
1answer
133 views

Chern-Simons and framing dependence$.$

According to ref.1, the correlation functions of a Chern-Simons theory are topological invariants, up to the so-called framing, that is, the trivialisation of $TM\oplus TM$. The origin of this framing ...
1
vote
5answers
584 views

Why do floating water drops form spheres?

Consider a drop of water floating in an inertial frame in STP air (e.g., the ISS). Intuitively, the equilibrium shape of the drop is a sphere. How would one prove that? Is it equivalent to showing ...