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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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95
votes
0answers
4k views

Linear sigma models and integrable systems

I'm a mathematician who recently became very interested in questions related to mathematical physics but somehow, I faced difficulties in penetrating the literature... I'd highly appreciate any help ...
40
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0answers
1k views

Can Lee-Yang zeros theorem account for triple point phase transition?

Now the prominent Lee-Yang theorem (or Physical Review 87, 410, 1952) has almost become a standard ingredient of any comprehensive statistical mechanics textbook. If the volume tends to infinity, ...
23
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0answers
326 views

Minimal strings and topological strings

In http://arxiv.org/abs/hep-th/0206255 Dijkgraaf and Vafa showed that the closed string partition function of the topological B-model on a Calabi-Yau of the form $uv-H(x,y)=0$ coincides with the free ...
19
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0answers
497 views

Hypersingular Boundary Operator in Physics

This has been a question I've been asking myself for quite some time now. Is there a physical Interpretation of the Hypersingular Boundary Operator? First, let me give some motivation why I think ...
18
votes
0answers
774 views

TQFTs and Feynman motives

Questions Is a topological quantum field theory metrizable? Or else a TQFT coming from a subfactor? For a given metric, are there always renormalization and Feynman diagrams? Is there always a Feynman ...
18
votes
0answers
854 views

p-Adic String Theory and the String-orientation of Topological Modular Forms (tmf)

I am going to ask a question, at the end below, on whether anyone has tried to make more explicit what should be a close relation between p-adic string theory and the refinement of the superstring ...
18
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0answers
709 views

Noether currents for the BRST tranformation of Yang-Mills fields

The Lagrangian of the Yang-Mills fields is given by $$ \mathcal{L}=-\frac{1}{4}(F^a_{\mu\nu})^2+\bar{\psi}(i\gamma^{\mu} D_{\mu}-m)\psi-\frac{1}{2\xi}(\partial\cdot A^a)^2+ \bar{c}^a(\partial\...
16
votes
1answer
530 views

Is there a physical interpretation to invariant random matrix ensembles?

Disclaimer. I am a graduate student in pure mathematics, so my knowledge of physics more advanced than basic 1st/2nd year undergraduate physics is very limited. I welcome corrections on any ...
15
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0answers
168 views

Is the converse of Weinberg's statement on the cluster decomposition principle true?

In Weinberg's "The Quantum Theory of Fields, Vol. 1", Section 4.4, page 182, the author says: We now ask, what sort of Hamiltonian will yield an $S$-matrix that satisfies the cluster decomposition ...
15
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0answers
456 views

Electric charges on compact four-manifolds

Textbook wisdom in electromagnetism tells you that there is no total electric charge on a compact manifold. For example, consider space-time of the form $\mathbb{R} \times M_3$ where the first factor ...
11
votes
0answers
361 views

Making new sense of the three-body problem in the light of Maryam Mirzakhani math contributions

I am unfamiliar with moduli spaces and ergodic theory which appear to be essential in Maryam Mirzakhani's math contributions which won her the Fields Medal. However, I am well conversant in essential ...
11
votes
1answer
1k views

String theory from a mathematical point of view

I have a great interest in the area of string theory, but since I am more focused on mathematics, I was wondering if there is any book out there that covers mathematical aspects of string theory. I ...
10
votes
0answers
219 views

Do the Wightman axioms uniquely fix the representation of the Poincaré group on the one-particle states given the representation on the fields?

Let $P := \mathrm{SL}(2,\mathbb{C})\ltimes \mathbb{R}^4$ be the universal cover of the connected component of the identity of the Poincaré group. Given a classical field $\phi : \mathbb{R}^{1,3}\to V$...
9
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0answers
317 views

Haag's theorem in a box

Haag's theorem states that the interaction picture does not exist in a rigorous way in relativistic quantum field theory. Some ways in which the interaction picture can fail are that the eigenstates ...
8
votes
0answers
167 views

Metric transformation, polygons and gravitons

I'm trying to understand the paper by Hitchin called: ''Polygons and gravitons". I'm stuck at page 471. At this point, he does some computations and obtains a metric: $$ \gamma dz d\bar{z}+\gamma^{...
8
votes
0answers
333 views

Objective time derivative that is not a Lie derivative

Summary Led by an interest into the concept of "Material Objectivity", I am asking myself: Are there objective time rates that are not Lie derivatives? The long read I am trying to understand the ...
8
votes
0answers
200 views

Wightman axioms always imply triviality in 4D?

Someone mentioned to me in passing that it had been proven that the Wightman axioms are over-restrictive in four dimensions and provably always result in trivial correlators (or maybe a trivial S-...
8
votes
0answers
424 views

Is the Hilbert space spanned by both bound and continuous hydrogen atom eigenfunctions?

As e.g. Griffiths says (p. 103, Introduction to Quantum Mechanics, 2nd ed.), if a spectrum of a linear operator is continuous, the eigenfunctions are not normalizable, therefore it has no ...
8
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0answers
191 views

Is there a null incomplete spacetime which is spacelike and timelike complete?

Geodesic completeness, the fact we can make the domain of the geodesic parametrized with respect an affine parameter the whole real line, is an important concept in GR. Especially, because the lack of ...
8
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0answers
182 views

Topology-dependent groud state degeneracy of $B \wedge F + B \wedge B$ and $B \wedge F + B \wedge B \wedge B$

There are some examples of topological BF theory with extra terms allow it still being topological. See this Ref. paper In 4d (3+1D), we have the trace of: $$ \int\frac{k}{2\pi}\text{Tr}[B \wedge F + ...
8
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0answers
264 views

Coleman-Mandula theorem in mathematical language

Every supersymmetry text starts off mentioning the Coleman-Mandula theorem. Often it is introduced using rather colloquial terminology. I was wondering if anyone knew a precise mathematical ...
8
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0answers
212 views

The Integral Trick and An Equality in Nakajima's Lecture

In Nekrasov et al's series papers MNS, they calculate such kinds of integral $$ \frac{E_1 E_2}{N(2\pi i)^N(E_1+E_2) }\oint d\phi_1 \wedge d\phi_2\wedge \ldots\wedge d\phi_N \prod_{i<N} (-\phi_i) \...
7
votes
0answers
70 views

How can we deduce that a hydrogen atom is stable in relativistic QED?

Consider relativistic quantum electrodynamics (QED) with three quantum fields: the electromagnetic field $A_\mu$, one fermion field $\psi$ for electrons/positrons, and one fermion field $\psi'$ for ...
7
votes
0answers
394 views

Level-rank duality in WZW models and CS theories

Cross-posting from Physics Overflow: https://www.physicsoverflow.org/41281/level-rank-duality-in-wzw-models-and-cs-theories I know that the classical level-rank duality in the $\widehat{\mathfrak{sl}}...
7
votes
0answers
62 views

When does the correlator of a string of fields and the current vanish “sufficiently fast” at infinity and Ward's identity?

One consequence of the Ward identity (cf. Di Francesco et al) is that it means variation of correlators under infinitesimal transformation is zero. This can be seen by integrating the ward identity, ...
7
votes
2answers
415 views

Intuitively what's the relationship between forces and connections?

In Einstein's General Relativity we relate the effects of gravity with the curvature of the Levi-Civita connection on the spacetime manifold. Also, when we get the electromagnetic tensor $F = dA$ ...
7
votes
0answers
372 views

1-form formulation of quantized electromagnetism

In a perpetual round of reformulations, I've put quantized electromagnetism into a 1-form notation. I'm looking for references that do anything similar, both to avoid reinventing the wheel and perhaps ...
6
votes
0answers
95 views

How are local observables encoded in this formulation of quantum field theory as a functor?

I've recently begun trying to understand a formulation of quantum field theory as a functor from a category of spacetimes-with-boundaries (bordisms) to a category of Hilbert spaces, as reviewed in [1]....
6
votes
1answer
155 views

What classifies gaugings?

Gauging a global symmetry $G$ introduces several free parameters to the theory. For example, In $d=4$, gauging a simple and simply-connected Lie group introduces a coupling constant and a theta term, ...
6
votes
0answers
159 views

What is the physical interpretation of the automorphism on bounded operators induced by an S matrix?

In a QFT, the S-matrix $S$ is a unitary operator, that fixes the vacuum and commutes with the unitary operators implementing the action of the Poincare group on an appropriate Hilbert space $H$. ...
6
votes
0answers
905 views

What is a modular tensor category / functor?

I have reads several answers here about this notion, especially regarding topological order, see e.g. this answer, but this notion sounds completely new for me. Also, I found nothing really helpful on ...
6
votes
0answers
247 views

Level quantization of 7d $SO(N)$ Chern-Simons action

In 3d, one can write down the $SO(N)$ Chern-Simons action to be $$S(A)=\frac{k}{192\pi}\int_{M}\text{Tr}(A d A +\frac{2}{3}A^3),$$ where $A$ is an $SO(N)$ connection. The level quantization can be ...
6
votes
0answers
329 views

explicit matrix elements for a representation decomposed into subgroup by branching rules

I'm looking for a way to construct a representation for a simple Lie group such that one particular subgroup is manifest. I learned the branching rules from Cahn, Georgi and Slansky, but I'm still not ...
6
votes
0answers
254 views

Physical interpretation: weighted eigenvalues of the Laplacian with a potential

I'm a mathematician with only the basic knowledge of Physics, so my question may be trivial: in this case, mercy me. :-) Let $\Omega \subseteq \mathbb{R}^N$ be a domain and let $V,m:\Omega \to \...
6
votes
0answers
67 views

What is the importance of studying degeneration on $M_g$

Let $M_g$ be the moduli space of smooth curves of genus $g$. Let $\overline{M_g}$ be its compactification; the moduli space of stable curves of genus $g$. It seems to be important in physics to study ...
5
votes
0answers
67 views

Physicist path integral and cylinder set measures

Path integral via discretization So let me start with what seems to be the point of view of physicists (corrections are highly appreciated since this is what I understood!). Let a quantum system with ...
5
votes
0answers
99 views

Intuitive/Physical reason why fields are distributions

I read in Urs Schreiber's notes on mathematical QFT that the infinities in the standard approach to QFT appear because the product between operator-valued field distributions is not always well ...
5
votes
0answers
376 views

Are there any open problems in mathematics whose resolutions would have important physical implications?

Are there any open problems in pure mathematics whose resolutions would resolve a well-formulated open physics question, whose solution could (at least in principle) explain an experimentally ...
5
votes
0answers
181 views

How can I get the Seiberg-Witten curve from M-theory?

I know that we can use the SU($n$) 6d (2,0) SCFT, the M5-brane world-volume theory to get $\mathcal{N}=2$ theories. Still, e.g. reading Tachikawa's "Supersymmetric dynamics for pedestrians" I cannot ...
5
votes
0answers
391 views

Is there a Virasoro group?

On page 14 of the survey article Kac-Moody and Virasoro algebras in relation to quantum physics by Goddard and Olive, the authors show that smooth selfmaps of the circle form a Lie group corresponding ...
5
votes
0answers
113 views

BF theory without perturbation theory (spin network appoach)

The BF model given by the action $$S = \int \mathrm{tr}(B \wedge F)$$ with a 2-form field $B$ and fibre bundle curvature tensor $F$ will have the following partition function after integration over $...
5
votes
0answers
185 views

What are the applications of hyperbolic $3$-manifold theory to cosmology?

I am a pure mathematician specialized in hyperbolic $3$-manifold topology. That has been an incredibly active field of research in the past few decades due to the seminal work of Thurston, as many of ...
5
votes
2answers
396 views

Combining rotations for orbital and rotational motion of a planet

If I have a planet orbiting the Sun (assuming circular orbit) at angular velocity $\Omega$ and rotating about its axis at $\omega$. I also have a normal to the surface of the planet $\vec{n}_{\rm surf}...
5
votes
0answers
135 views

Is there an algorithm to diagonalize a matrix using gauge transformations

I have two matrices $U(\lambda, x,t)$ and $V(\lambda, x,t)$, where $\lambda$ is a parameter, which belong to the $sl(2)$ algebra, and satisfy the zero-curvature equation $$ \partial_t U - \partial_x V ...
5
votes
0answers
352 views

How does one determine if a spacetime is globally hyperbolic?

A spacetime $M$ is said to be globally hyperbolic if it is strongly causal and if the sets $J^+(p)\cap J^-(q)$, for all $p,q\in M$, are compact. (For more information, see the Wiki article on causal ...
5
votes
0answers
335 views

Free probability in Physics

Recently I have started reading some materials on non-commutative probability. IN this area mathematicians sometimes consider quantum theory as a non-commutative version of classical probability, with ...
5
votes
0answers
411 views

Monstrous Moonshine outside of String Theory

My question concerns applications of monstrous moonshine, which is the connection between the $j$-function and the monster group. Recently, physicists have applied it to string theory and, ultimately, ...
5
votes
0answers
295 views

Topological Quantum Field Theories

I've asked this on Math.SE, but with no avail. So, I decided to ask it here. I was wondering about the following after reading the Wikipedia article on TQFTs. It is said that TQFTs have vanishing ...
5
votes
0answers
182 views

Understanding the intermediate field method for the $\phi^4$ interaction

In Rivasseau's and Wang's How to Resum Feynman Graphs, on page 11 they illustrate the intermediate field method for the $\phi^4$ interaction and represent Feynman graphs as ribbon graphs. I had to ...
5
votes
0answers
420 views

What are endomorphism bundle valued $p$-forms and exterior covariant derivatives and their use in Chern-Simons theory?

Chern-Simons Forms appears in several places in physics for examples, Fractional Quantum Hall Effect, response of Topological Insulator, invariant of knot, electromagnetism in 2+1 space-time, and ...