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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38 views

Mass terms in the SUSY gauged linear sigma model

Okay, I have a very basic question about the SUSY gauged linear sigma model which is driving me crazy. I am following Chapter $15$ of Mirror Symmetry by Hori et al. I am considering the SUSY gauged ...
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Anticommutation of variation $\delta$ and differential $d$

In Quantum Fields and Strings: A Course for Mathematicians, it is said that variation $\delta$ and differential $d$ anticommute (this is only classical mechanics), which is very strange to me. This is ...
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With a local anomaly, is the determinant of the Dirac operator still a section of a complex line bundle?

In the literature about anomalies in quantum field theory, the determinant of the Dirac operator plays an important role. The Dirac operator may depend on some background data, and the subject of ...
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Coleman–Mandula theorem and its assumptions on QFT

The description of the Coleman–Mandula theorem on Wikipedia starts with the following assumptions: Every quantum field theory satisfying the assumptions, Below any mass M, there is only a finite ...
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1answer
32 views

How to take trace over group and Dirac indices?

I'm currently reading Pokorski's book "Gauge Field Theories" and in Chapter 13 he discusses, among other things, Fujikawa's method of deriving the chiral current (see page 488 and the ...
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1answer
59 views

As a physicist, why are associated bundles important?

I have a good grasp on principal bundles as providing a lie group on some fibers of our field. So for example, the wavefunction tells us the phase of a particle in space and time, and this can be a ...
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148 views

Is mathematical physics theoretical physics? [duplicate]

I was reading about mathematical physics on a university website and they said "an education in theoretical physics is..." appearing to imply mathematical physics is theoretical physics. Is ...
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58 views

Relation between self-adjointness and variational principle and Rayleigh's principle

In mathematical physics, why is it that when an eigen-equation is described by a self-adjoint operator we say that it can be written (formulated) as a variational action (or principle)? Does the ...
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73 views

When and why can the spin connection term of the Dirac Operator be omitted?

The Dirac Operator $D$ is defined by \begin{equation}\tag{1} D=i\gamma^a\nabla_a=i\gamma^a\nabla_{e_a}=i\underbrace{\gamma^a{e_a}^\mu}_{=\gamma^\mu}\nabla_{\partial_\mu}=i\gamma^\mu\nabla_\mu=i\gamma^\...
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1answer
49 views

Hyperbolic isometries in the context of General Relativity

In the context of hyperbolic geometry, it is possible to create a classification for isometries. I would like to know if these isometries have any particular meaning in the context of general ...
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Is there any physical significance for the Cremona group / birrational transformations on the projective plane?

In algebraic geometry, the Cremona group is the group of birational transformations on the projective plane. I would like to know: Is there any physical interpretation for this group? Does this group ...
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1answer
37 views

Does a von Neumann algebra correspond to a quantum system?

We usually associate a quantum system with a Hilbert space $\mathcal{H}$ and consider the set of bounded operators on $\mathcal{B}(\mathcal{H})$. Especially the set of unit-trace and positive (semi-...
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Can a QFT be anomaly-free on spacetimes that are boundaries but still have an anomaly on other spacetimes?

If $D$ is the Dirac operator for some dynamic spinor fields in background gauge and gravitational fields, then the partition function is supposed to be $\mathrm{det}(D)$. For this to make sense, we ...
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31 views

Why must an electric field be Fourier transformed to offer meaningful spectroscopic information?

I understand that it is the mathematical function needed to interpret the data, but this makes no sense to me mathematically. Why must an electric field (as a function of time) be Fourier transformed ...
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What is the physical importance of topological quantum field theory?

Apart from the fascinating mathematics of TQFTs, is there any reason that can convince a theoretical physicist to invest time and energy in it? What are/would be the implications of TQFTs? I mean is ...
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Anomalies in QFT: why do we require smooth dependence on the background fields?

If $D$ is the Dirac operator for some dynamic spinor fields in a background gauge field $A$, then the partition function is supposed to be $\mathrm{det}(D)$. But if the coupling to the gauge field is ...
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122 views

How to define a map from $\mathrm{SO}^+(1, 3)$ to $\mathrm{SL}(2, \mathbb{C})$?

To explain how spinors transform under rotation people often use that $\mathrm{SL}(2, \mathbb{C})$ is a double cover $\mathrm{SO}^+(1, 3)$, that is there is a surjective homomorphism $\lambda:\...
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141 views

Why do you need to count curves in string theory?

One of the mathematical fields that string theory is said to have had a large bearing on is enumerative geometry which, roughly, deals with counting rational curves on hypersurfaces and its ...
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109 views

Particle in a box with absolutely continuous spectrum

Let's consider a particle on a real line in a potential $V(x)$ which disappears at infinity. The Hamiltonian is: $$ H: W^{2,2}(\mathbb{R}) \subset L^2(\mathbb{R}) \to L^2(\mathbb{R}) \\ \big( H \, ...
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Are there any results about regularity of the generalized eigenfunctions used in spectral representations of unbounded differential operators?

I am using the "Direct integral" version of the spectral theorem, as given e.g. in Hall. It says that a diagonal representation for an unbounded operator on the Hilbert space $L^2(\mathbb{R}^...
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10 views

Helmholtz decomposition of magnetic field generated by a infinitely-long line charge in uniform axial motion

Context In [1], I derived the magnetic field generated by a infinitely-long line charge that is in uniform motion in a direction co-linear to the line of charge. The method that I used was volume ...
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162 views

Fermat's Last Theorem in Physics

I was wondering if Fermat's Last Theorem can relate somehow to quantum numbers and energy spectrum in some theoretical system. Are there any examples for such systems? And in general, Is there any use ...
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64 views

Bounding the value of a function for separable spin states

Consider $N$ spin-1/2, for which we can define the collective spin operator $\vec{S}=\sum_i \frac{\vec{\sigma}^{(i)}}{2}$. My question is, what is the upper bound $U$ on $$ f(\rho) = \text{Var}[ S_z ] ...
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Is the particle-hole transform rigorous in infinite-dims?

$\DeclareMathOperator{\tr}{tr}$Let me only consider fermions, i.e., fermionic Fock space $\mathscr{F}$ of a single-particle Hilbert space $\mathscr{H}$. Let us assume that $\dim \mathscr{H}=\aleph_0$ (...
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76 views

Instantons in Minkowski spacetime? or only valid in Euclidean spacetime?

In the usual description of the instanton of nonabelian gauge theory in $D=4$ spacetime, we always (or just usually?) choose the $D=4$ Euclidean spacetime see for example https://en.wikipedia.org/wiki/...
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104 views

How do *-Algebras correspond to operators on a Hilbert space?

In algebraic quantum field theory, a theory is defined through a net of observables $\mathcal{O} \mapsto \mathcal{A}(\mathcal{O})$ fulfilling the Haag-Kastler axioms (see e.g. this introduction, sec. ...
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53 views

How to Wick rotate the Yang-Mills instanton winding number?

How to Wick rotate the instanton number of Yang-Mills theory? (Related to the earlier question Wick rotate the Yang-Mills $SU(N)$ gauge theory's field strength?) My question is particularly about ...
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27 views

If having a finite action for Maxwell or Yang-Mills theory, $A_μ$ goes to a pure gauge configuration?

I had an empirical understanding that --- If we like to have a finite action for Maxwell or Yang-Mills theory, so that the field strengh $F_{μν}$ must go to zero at space-time infinity, meaning that $...
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The sign of axion $F$ $F$ dual term in Weinberg

Related to the earlier question $\gamma^5$ rotation of chiral fermion in (1) Peskin&Schroeder, (2) Weinberg, or (3) Srednicki. The sign of axion $F$ $F$ dual term in Weinberg (23.6.16) appears to ...
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Are there sub-exponential local complex partition functions?

Consider an arbitrary local, translation-invariant, classical statistical lattice model such as the Ising or Potts model. The partition function $Z$ is a sum over products of local Boltzmann weights, ...
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19 views

What is the 'free' form or 'free' version of an equation in mathematical physics? Like the Dirac equation, (among others)

From Wikipedia: In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it ...
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When to use (and when not to use) electromagnetic field conjugates in variational formulations

I found something a little bit confusing about writing variational formulas or Lagrangians for electromagnetic fields. I was looking at the book by Schwinger and Milton (chapter 4), and saw that ...
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What is the equation for calculating magnetic field for two coils when the currents in the coils have opposite orientation?

So I was wondering what the equation is to calculate the magnetic field when the two coils are separated by a distance d, with the currents in the coils in opposite orientation? Any help would be ...
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33 views

Electric and magnetic charge quantization: other derivations or perspectives?

We know (say from Griffiths E&M Problem 8.12) that the electric $q_e$ and magnetic charge $q_m$ (with a distance $\vec{z}$ apart) can store the angular momentum in the space: $$ \vec{L}=\int d^3 V ...
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23 views

Good books on Green's functions [duplicate]

Could anyone recommend introductory books on Green's functions with applications in the framework of classical electrodynamics?
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16 views

Partition functions of descendent SPTs of the Haldane chain

The Haldane chain can be viewed as a $1+1$ D SPT protected by an $SO(3)$ symmetry. If this SPT is put on a triangulated closed manifold $X$, its partition function can be written as $$ e^{i\pi\...
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Relation between the boundedness and discreteness of conjugate operators

I have two very general questions about operators in quantum mechanics. Suppose $A$ and $B$ are self-adjoint operators associated to conjugate physical quantities (e.g. position/momentum), meaning ...
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102 views

D'Alembertian of a delta-function of a space-time interval (i.e. on the light-cone)

How one differentiates a delta-function of a space-time interval? Namely, $$[\partial_t^2 - \partial_x^2 - \partial_y^2 - \partial_z^2] \, \delta(t^2-x^2-y^2-z^2) \, .$$ Somewhere I saw that the ...
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If $f(m_1,m_2-1)=f(m_1-1,m_2)$, does it mean that $f(m_1,m_2) $ is independent of $m_1$ and $m_2$?

In this article Theory of Complex Spectra. II Giulio Racah defines $f(m_{1} m_{2} ; jm)$ by $$ \left(m_{1} m_{2} \mid j m\right)=(-1)^{j_{1}-m_{1}} f\left(m_{1} m_{2} ; j m\right)\left[\left(j_{1}+m_{...
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Why does a choice of this $\psi$ in the worldsheet metric corresponds to a choice of complex structure?

As far as I'm aware, a complex manifold $M$ is a topological manifold together with an atlas ${\cal A}$ of charts $(U_i,\varphi_i)\in{\cal A}$ such that the open sets $U_i$ cover $M$, the maps $\...
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94 views

Where do fermionic coherent states live?

Although there have been a couple of questions on fermionic coherent states, I don't think any has answered the question "on what space do fermionic coherent states live?", or at least not ...
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211 views

Understanding Hamilton's equations in classical field theory in a rigorous way

So, I'm in a quest of understanding classical field theory on my own, and I'm interested in its rigorous construction. Here's the link for a previous post of mine on mathoverflow. The interesting ...
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103 views

Rigorous delta potential – a formulation using distributions?

It is common in QM, even in mathematical physics, to consider Hamiltonians with a Dirac delta in the potential: $$ \hat H = -\frac{\mathrm{d}^2}{\mathrm{d}x^2} + V(x) + \delta(x-a) \: . $$ The most ...
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1answer
76 views

How these two approaches to spinors in curved spacetimes relate?

Regarding spinors in curved spacetimes I have seem basically two approaches. In a set of lecture notes by a Physicist at my department he works with spinors in a curved spacetime $(M,g)$ by picking a ...
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109 views

Riesz Representation Theorem and Inner Products

I am running into some confusion: Suppose in Quantum Mechanics we think of the ket $|x \rangle $ as being an improper state - one not "actually" in the Hilbert space, but which is useful to ...
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1answer
60 views

Why is locally synchronizable defined as $\omega\wedge \mathrm d\omega=0$?

I'm reading GTM48: General relativity for mathematicians, and I have a problem about the definition of locally synchronizable, which is defined as $\omega\wedge \mathrm d\omega=0$. Let $(M,g,D)$ be a ...
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98 views

What is an intuitive way of understanding Ostrogradsky instability?

Is there a way of explaining why no differential equations in physics exceed order two without delving into Lagrangian and Hamiltonian mechanics - i.e. from Newtonian mechanics? Moreover, is there ...
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30 views

Which non-homogeneous scalar manifolds are possible in supergravity?

The scalar fields in various supergravity theories are restricted (by supersymmetry) to span the target (scalar) manifolds of a certain class (e.g. Hodge-Kahler, quaternion-Kahler etc.), depending on ...
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21 views

Hypersurface Orthogonality of Conformal Killing field on Conformal Killing Horizon

Suppose we have a spacetime manifold $(\mathcal{M},g)$ admitting a conformal Killing Horizon $\mathcal{H}_c$ generated by a conformal Killing field $\chi^a$ (which is null only on the conformal ...
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77 views

Why to introduce spinor fields we need this map in the definition of a spin structure?

Let me start with what I currently understand. Let ${\rm SO}(1,3)$ be the proper ortochronous Lorentz group. Its universal cover is ${\rm SL}(2,\mathbb{C})$. The representations of its universal cover ...

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