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3
votes
1answer
281 views

Susy QM and Atiyah-Singer index theorem

Consider maps $t\mapsto x^i(t)$ from circle to some Riemannian (spin) manifold and lagrangian $$ \mathcal L = \frac12 g_{ij}(x) \partial_t x^i \partial_t x^j + \frac12 g_{ij} \psi^j \left(\delta^i_k ...
5
votes
2answers
394 views

Is there a 1-1 correspondence between symmetry and group theory?

The professor in my class of mathematical physics introduces the definition of groups and said that group theory is the mathematics of symmetry. He gave also some examples of groups such as the set ...
0
votes
0answers
56 views

What lives in the Hilbert Space? [duplicate]

Consider the eigenvalue equation: $$\hat{Q}\Psi = q\Psi$$ where $q$ and $\Psi$ are eigenvalues and eigenfunctions of the hermitian operator $\hat{Q}$. If the spectrum of the hermitian operator is ...
12
votes
2answers
461 views

Proof of Loss of Lorentz Invariance in Finite Temperature Quantum Field Theory

In the standard quantum field theory we always take the vacuum to be a invariant under Lorentz transformation. For simple cases, at least for free fields, is very simple to actually prove this. Now ...
2
votes
0answers
42 views

Mathematical text for Astrophysics [closed]

Please recommend me some Mathematical text books and a list of Mathematical topics for Astrophysics.
3
votes
2answers
228 views

Proof of quantization of magnetic charge of monopoles using homotopy groups

Suppose we place a monopole at the origin $\{{\bf 0}\}$, and the gauge field is well-definded in region $\mathbb R^3-\{0\}$ which is homomorphic to a sphere $S^2$. Then the total manifold is $U(1)$ ...
7
votes
3answers
388 views

Is the harmonic oscillator potential unique in having equally spaced discrete energy levels?

I was wondering if the good old quadratic potential was the only potential with equally spaced eigenvalues. Obviously you can construct others, such as a potential that is infinite in some places and ...
4
votes
0answers
112 views

References to Mechanics (Classical, Quantum, Statistical) using Time-Scale calculus?

Time-Scale Calculus, is a theory which unifies ordinary (plus fractional and q-) calculus with discrete (and finite differences) calculus. In a sense, in a similar way the Lebesgue integral (or ...
2
votes
0answers
121 views

Geometric interpretation of Grassmann variable

Grassmann variables were introduced to make path-integral formalism easier to handle fermionic (anti-commutating) fields. Mathematically they represent the exterior algebra of forms (or exterior ...
1
vote
0answers
58 views

Any physical expressions using the integral of the Bessel function of the Second Kind? [closed]

I am currently studying the asymptotic expansions of the integral of the form $$ I(b)=\int_{0}^{\infty} \phi (x) Y_ \nu (bx) dx $$ where $ \phi $ is an element of Schwartz space and $Y_ \nu $ is the ...
1
vote
0answers
49 views

What is a Chiral Algebra for a group?

What do we mean by the Chiral Algebra for a group G (SO(3) etc )? Do you know a reference suitable for physicists? Thank you
1
vote
0answers
86 views

Is Functional Analysis used at all in String Theory? [closed]

Is functional analysis used at all in string theory? (and if so, how?)
0
votes
0answers
73 views

Robin boundary conditions for the heat equation

Consider a 3D channel with fluid or gas with walls $\Gamma_1$, inflow part $\Gamma_2$ and outflow part $\Gamma_3$. The temperature is described by the heat equation: $$ \frac{\partial T}{\partial t} ...
1
vote
1answer
129 views

Is the scattering length definitely positive if the potential is everywhere nonnegative?

Is the scattering length definitely positive if the potential is everywhere nonnegative? Intuitively, it seems reasonable. Any rigorous proof?
4
votes
1answer
156 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 ...
2
votes
1answer
100 views

Superfluid rotating frame of reference

I'm currently studying a text about Bose-Einstein condensates (BECs) and vortices. When they want to study whether a vortex will be formed, they look at the fact wether it's enegetically favorable. ...
7
votes
2answers
291 views

Interpreting some domain issues of (potential) momentum operators

In the context of mathematical quantum mechanics, a well known no-go theorem known as Hellinger-Töplitz tells us that an unbounded, symmetric operator cannot be defined everywhere on the Hilbert space ...
2
votes
3answers
124 views

How do the flow equations relate to the actual situation?

This question might seem silly but I'll try to make it clear. It's a question (I think) about partial differential equations systems in general, but since currently I'm studying fluid mechanics I'll ...
2
votes
0answers
22 views

Main differences between elastodynamic and light scattering when using S-matrix to find bound states

What are the main differences (top 5 if question is too broad), for using the S-matrix to find bound states, between elastodynamic and light scattering? (if it facilitates a higher quality ...
8
votes
2answers
3k views

Differences between symmetric, Hermitian, self-adjoint, and essentially self-adjoint operators

I am a physicist. I always heard physicists used the terminology "symmetric", "Hermitian", "self-adjoint", and "essentially self-adjoint" operators interchangeably. Actually what is the difference ...
10
votes
4answers
521 views

Electromagnetic field and continuous and differentiable vector fields

We have notions of derivative for a continuous and differentiable vector fields. The operations like curl,divergence etc. have well defined precise notions for these fields. We know electrostatic and ...
0
votes
0answers
52 views

Action principles and covariant equations [duplicate]

Can every physically sound differential equation, that is covariant, deterministic etc. be derived by extremising a suitable action using a suitable lagrangian, that may be arbitary. Is this a ...
7
votes
1answer
538 views

Hilbert space of a free particle: Countable or Uncountable?

This is obviously a follow on question to the Phys.SE post Hilbert space of harmonic oscillator: Countable vs uncountable? So I thought that the Hilbert space of a bound electron is countable, but ...
6
votes
0answers
96 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 ...
7
votes
3answers
767 views

Mathematical understanding of Quantum Mechanics

Assuming that $\phi(r) = F (\psi(r))$ for some operator $F$ in Quantum Mechanics. Then, in our lecture today, we said that $$\phi(r) = \langle r|F |\psi\rangle = \int_{\mathbb{R}} \langle r |F| r' ...
8
votes
2answers
2k views

Book covering Topology required for physics and applications

I am a physics undergrad, and interested to learn Topology so far as it has use in Physics. Currently I am trying to study Topological solitons but bogged down by some topological concepts. I am not ...
4
votes
3answers
1k views

Why does Law of Large Numbers work?

Often I see books and professors reasoning that, in order to make a good experiment, many measurements are necessary because then the average value of a quantity is closer to the expected value ...
2
votes
1answer
97 views

product solutions for PDEs, physical motivation

Given a boundary value problem with independent variables $x_1,x_2, \dots , x_n$ and a PDE say $U(x_i, y, \partial_j y,\partial_{ij} y, \dots )=0$ we typically begin constructing a general solution by ...
3
votes
2answers
177 views

Domain of simple quantum harmonic oscillator

When discussing the spectral theory of unbounded operators, one often starts with an operator defined on a densely defined subspace of your Hilbert space, and then proves that the operator is ...
1
vote
1answer
60 views

Correspondence between one-parameter subgroups of $G$ and $T_eG$

I am reading the proof of this theorem from Andreas Arvanitoyeorgos and I cannot get some points in it, highlighted below. Theorem. The map $\phi \to d\phi_0(1)$ defines a one-to-one correspondence ...
11
votes
0answers
352 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 ...
2
votes
0answers
138 views

Several Complex Variables in QFT

After reading the very interesting quote about several complex variables in QFT: "The axiomatization of quantum field theory consists in a number of general principles, the most important of ...
2
votes
0answers
104 views

Learning Roadmap to Mathematical Physics [duplicate]

Currently, I am a graduate student specializing in algebraic geometry. On the other hand, I have also become extremely interested in the mathematical physics. However, I am not sure what steps I ...
3
votes
0answers
84 views

Hopf Algebras in Quantum Groups

In the theory of quantum groups Hopf algebras arise via the Fourier transform: A third point of view is that Hopf algebras are the next simplest category after Abelian groups admitting Fourier ...
1
vote
1answer
114 views

Momentum Representation vs Position Representation

We are given an operator $g$ from $\mathcal{l}^2(\mathbb{Z})$ to $\mathcal{l}^2(\mathbb{Z})$, i.e., the space of functions that are square summable over $\mathbb{Z}$ such that ...
2
votes
1answer
225 views

How to choose the Correct Green's Function?

In order to solve the Green’s function of the Helmholtz operator $$(\nabla^2+k^2)G(\vec r-\vec r’)=\delta^{(3)} (\vec r-\vec r’)$$ one can obtain four different Green’s functions corresponding to four ...
3
votes
1answer
360 views

Paths in the path integral

In the path integral approach one defines in some heuristic way the functional path integral \begin{equation} Z=\int{\cal{D}}\phi e^{iS(\phi)} \end{equation} and the one claims that one must ...
4
votes
1answer
311 views

Research problems in application of Lie groups to differential equations [closed]

Are there any open problems in physics involving Lie groups and differential equations for a phd theses. Some applications are say, Noether's theorem in classical or quantum field theory. But I am ...
11
votes
4answers
761 views

Complex integration by shifting the contour

In section 12.11 of Jackson's Classical Electrodynamics, he evaluates an integral involved in the Green function solution to the 4-potential wave equation. Here it is: $$\int_{-\infty}^\infty dk_0 ...
1
vote
1answer
55 views

Finding expectation value of $p^2$ without integrals

So the expectation value of momentum, if you know the expectation value of position is $$\langle p \rangle = m \frac{d\langle x \rangle}{dt}$$ Is there a nice formula like this for $\langle p^2 ...
0
votes
1answer
146 views

Which textbooks contain info on Bessel functions & their use as basis functions?

As an exercise my research mentor assigned me to solve the following set of equations for the constants $a$, $b$, and $c$ at the bottom. The function $f(r)$ should be a basis function for a ...
7
votes
0answers
121 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 ...
1
vote
1answer
94 views

Proof oriented subjects, similar to computational complexity [closed]

I'm starting my second year as an undergrad math major. I quite like the kind of thought involved in my pure math classes (analysis, abstract algebra), but I also like my physics and (theoretical) ...
1
vote
2answers
72 views

Could a trial wavefunction providing exact eigenenergy differ from the exact eigenfunction by a zero measure function?

Given the eigenequation of a Hamiltonian $$ H |n \rangle = E_n |n \rangle \tag{1} $$ We write it in the position representation $$ \langle x | H | n \rangle = E_n \langle x | n \rangle \tag{2} $$ ...
17
votes
4answers
2k views

Applications of the Spectral Theorem to Quantum Mechanics

I'm currently learning some basic functional analysis. Yesterday I arrived at the spectral theorem of self-adjoint operators. I've heard that this theorem has lots of applications in Quantum ...
2
votes
2answers
191 views

What makes an abstract physical system describable by a “fluid” equations of motion?

We can describe (some of) the dynamics of many systems using fluid mechanics. Of course these include classical fluids like water, more exotic fluids like photon gases and the universe as a whole and ...
5
votes
1answer
62 views

What's the definition of incompleteness of a coordinate system and a spacetime?

I always see in GR textbooks that some coordinates or some spacetime is incomplete, such as Rindler spacetime and spacetially flat FRW universe with only positive cosmological constant. This ...
1
vote
2answers
221 views

Can we describe Quantum Mechanics using filters and matrices? [closed]

Can mathematical filters or ultrafilters be used to predict quantum physics 'events' as accurately as using matrices like Schrodinger did? Is there a way to explain some of the predictive power of ...
7
votes
1answer
152 views

Self-adjointness

I know I have posted this question before some time ago. But no one could help so I decided to put my problem in another background. The Schrödinger equation of a free scalar field is given by ...
1
vote
0answers
101 views

Disclinations, dislocations, lattices, Displacement fields and scaling

I am looking up Frank, and Burger vectors and associated material on dislocation/disclination. It seems straightforward describing a lattice and what dislocation means. It is even possible to restrict ...