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 ...

learn more… | top users | synonyms

0
votes
1answer
45 views

What are absolutely continuous spectrum and singularly continuous spectrum?

I am now reading some mathematical note on Anderson localization. It mentioned two types of continuous spectrum. What are absolutely continuous spectrum and singularly continuous spectrum? I only had ...
3
votes
3answers
313 views

Does the variation of the Lagrangian satisfy the product rule and chain rule of the derivative?

I have seen wikipedia use the product rule and maybe the chain rule for the variation of the Langragin as follows: \begin{align} \dfrac{\delta [f(g(x,\dot{x}))h(x,\dot{x})] } {\delta x} = \left( ...
1
vote
3answers
77 views

What is the relationship between $V(t)$ and $V(x,y,z)$

I was recently asked this by a friend. He told me that coming from a physics background, he does not understand $V(t)$ and he believes it is purely theoretical construct made up by circuit ...
3
votes
1answer
152 views

Graph Theory and Feynman Integrals

In Vladimir A. Smirnov's book Analytic Tools for Feynman Integrals, Section 2.3, the alpha representation of general Feynman integral takes the form $$ F_{\Gamma}(q_1,\ldots,q_n;d) = ...
7
votes
2answers
214 views

Mathematical physics text with plenty of applications

I'm looking for texts on mathematical physics. I've seen various other threads, but the texts recommended in those threads were mathematical methods of theoretical physics texts, that is to say those ...
2
votes
1answer
67 views

Approach to expressing $|n\rangle\langle n| $ as a polynomial when eigenvalues are degenerate?

If ${|n\rangle}$ are eigenvectors of an operator $A$ then $|n\rangle\langle n| $ can be expressed in terms of a finite order polynomial $$|n\rangle\langle n| =\prod_{m\ne n} \frac{A-a_m}{a_n-a_m}$$ ...
6
votes
1answer
218 views

Mathematical interpretation of Poisson Brackets

Lets say we are working in a classical scalar field theory and we have two functional $ F[\phi, \pi](x)$ and $G[\phi, \pi](x)$. In most of the references, starting with two functional the Poisson ...
2
votes
1answer
88 views

Is the fine structure constant a rational number?

Since the fine structure constant (denoted alpha) is a pure real number, it just occured to me to ask if it is a rational number or not.
1
vote
1answer
96 views

Moduli spaces in string theory vs. soliton theory

In both string theory and soliton theory, moduli spaces are frequently used. As far as I known, for soliton theory, moduli spaces are something like collective coordinates for solitons, and for ...
0
votes
2answers
94 views

A new way of looking and formulation of Observables for a new quantum theory

I often think of basic of QM (although it wasnt discovered that way) is that we have a physical parameter/observable and the favourite one is the displacement $x$ of a particle $P$. Its conjugate ...
4
votes
3answers
793 views

Intuitive meaning of the exponential form of an unitary operator in Quantum Mechanics

I'm an undergraduate student in Chemistry currently studying quantum mechanics and I have a problem with unitary transformations. Here in my book, it is stated that Every unitary operator ...
2
votes
0answers
66 views

What is elliptic genera?

What is elliptic genera in physics? Reading many relevant papers, they just defined elliptic genus as sort of partition function. I try to find useful materials to explain it, but I couldn't find ...
1
vote
1answer
78 views

The Euler-Poincare equation

Can anyone tell me very basically how the Euler-Poincare equation generalises the Euler-Lagrange equation? Further does anyone know if there is an "easy" relationship between the two, i.e. can anyone ...
6
votes
2answers
158 views

Eigenstates of a Hermitian field operator

Consider a Hermitian field operator $\phi(x)$ with eigenstates satisfying $$ \phi(x) |\alpha\rangle = \alpha(x) | \alpha \rangle $$ I'm trying to determine the inner product between the eigenstates. ...
2
votes
0answers
43 views

Mathematical books to become a successful mathematical physicists [duplicate]

My understanding of algebraic topology and Riemannian geometry come from Nakahara's Geometry, Topology, and Physics, which I do not think is sufficient. I am first year PhD student, and I want to do ...
1
vote
1answer
49 views

How do you handle a functional input in a Dirac delta function and prove these types of relations?

I have a quadratic relation inside of a Dirac delta function with the following relation \begin{align} \delta((x-x_1)(x-x_2)) = \dfrac{ \delta(x-x_1) + \delta(x-x_2) }{|x_1-x_2|}. \end{align} How do ...
2
votes
2answers
122 views

A step in zeta function regularization

I'm just wondering about the mathematical step $$\sum_{n=1}^\infty n\exp[-\epsilon n\sqrt x]=\frac1{\epsilon^2 x}-\frac1{12}+\mathcal O(\epsilon).$$ Why is this equality so? I see that ...
4
votes
1answer
226 views

How do you go from quantum electrodynamics to Maxwell's equations?

I've read and heard that quantum electrodynamics is more fundamental than maxwells equations. How do you go from quantum electrodynamics to Maxwell's equations?
5
votes
1answer
122 views

Transport theorem derivation question

I am trying to rigorously go through some fluid mechanics proofs and theorems. I am currently going through a proof related to the transport theorem and I am having trouble with a step. The steps in ...
15
votes
4answers
742 views

What is a good introduction to integrable models in physics?

I would be interested in a good mathematician-friendly introduction to integrable models in physics, either a book or expository article. Related MathOverflow question: what-is-an-integrable-system.
2
votes
1answer
90 views

Measurement of observables with continuous spectrum: State of the system afterwards

Suppose my system, described by a separable Hilbert space $H$, is in the state $\Psi$ when I measure an observable that has only continuous spectrum. What is the state of the system after the ...
1
vote
2answers
173 views

Gravity on a upward moving object

I know I should have listened to my teacher in physics class but I didn't. I need help on a thing I am trying to do for a game. I want to be able to calculate Gravity pushing down on a object over a ...
9
votes
4answers
993 views

What is the relation between (physicists) functional derivatives and Fréchet derivatives

I´m wondering how can one get to the definition of Functional Derivative found on most Quantum Field Theory books: $$\frac{\delta F[f(x)]}{\delta f(y) } = \lim_{\epsilon \rightarrow 0} ...
14
votes
5answers
688 views

Applications of Geometric Topology to Theoretical Physics

Geometric topology is the study of manifolds, maps between manifolds, and embeddings of manifolds in one another. Included in this sub-branch of Pure Mathematics; knot theory, homotopy, manifold ...
7
votes
2answers
522 views

Proof for the completeness of eigenfunctions of a self-adjoint operator

I always heard the eigenfunctions of a self-adjoint operator form a complete basis. Where can I find a proof in infinite dimension space? Presumably readable for physicists.
2
votes
1answer
78 views

Making an Incomplete Set of Observables Complete

In quantum mechanics, it seems a standard procedure that if you have an incomplete set of observables, then one can make this set complete by adding more commuting observables until the set becomes ...
3
votes
2answers
358 views

Question about infinite sum in quantum field

I read from some books of number theory that $$\sum_{n=1}^{\infty}\frac{1}{n^s} = -\frac{1}{12}\text{,when } s=-1.$$ Now is there such a result $$\sum_{n=1}^{\infty}\frac{1}{n^s} = \pi \text{,when } ...
2
votes
0answers
94 views

How to calculate the dispersion relation for a wave equation with non-constant speed of wave propagation?

Specifically, it is a one-dimensional wave equation for waves on a string with a non-constant cross-section, i. e. $$S(x)=S_1+S_2 \cos{2x}; \qquad c(x)=\sqrt{F/\rho\, S(x)}.$$ Separating the variables ...
1
vote
1answer
128 views

Null lines and degenerate plane

Can anyone explain me what null lines are and degenerate plane? I don't know anything about it, I don't have physics background and I am a mathematics student and please tell me if there is any good ...
4
votes
1answer
201 views

Wick Rotation in Curved space

So over time I have learned to do exhaustive searches before asking things here. Wick rotations are cool if you are trying to work in qft and make statements about the thermodynamics of some physical ...
1
vote
1answer
72 views

Hill's and Mathieu's equation [closed]

I am supposed to apply Hill's and Mathieu's equation to parametric pendulum. Can you tell me what is the difference between them? Why are they used? What do they describe?
4
votes
1answer
119 views

Is gauge connection unique?

In QFT, given a gauge group and matter field, is the form of the gauge field unique? In other words, given a principal G-bundle and its associated vector bundle, is the construction of the principle ...
3
votes
3answers
935 views

Can we have discontinuous wavefunctions in the Infinite Square well?

The energy eigenstates of the infinite square well problem look like the Fourier basis of L2 on the interval of the well. So then we should be able to for example make square waves that are an ...
1
vote
1answer
58 views

Prove the solution of von Neumann equation will never stabilize if Hamiltonian and initial density matrix commutes

Given von Neumann equation $$\frac{d}{dt} \rho(t) = -i [H, \rho(t)] = -i e^{-iHt}[H, \rho(0)]e^{iHt}.$$ If we know that $[H, \rho(0)] \neq 0$, how do we prove in details the solution of von Neumann ...
11
votes
3answers
390 views

What kind of manifold can be the phase space of a Hamiltonian system?

Of course it should have dimension $2n$. But any more conditions? For example, can a genus-2 surface be the phase space of a Hamiltonian system?
5
votes
1answer
200 views

Hilbert space for Density Operators (instead of Banach spaces)

Is it possible to construct a well defined inner-product (and therefore orthonormality) within the set of self-adjoint trace-class linear operators? In the affirmative case, dynamics could be analyzed ...
22
votes
6answers
3k views

Are there two theories that are mathematically identical but ontologically different?

I have no background in physics but there is a question that has been bothering me, so I'm asking you. Are there at least 2 physical theories that are : Mathematically identical, which means that ...
18
votes
2answers
2k views

What interesting physics problems can't be solved because mathematics is not developed enough? [closed]

I'm curious as to what sorts of physical problems to which we don't have an answer, because we haven't developed the right mathematics yet (or advanced-enough mathematics). Related to this question ...
9
votes
0answers
148 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 ...
1
vote
0answers
47 views

How can one diagonalize the second variation of action?

Suppose we have action $S[q]$ and its stationary path $q_s$, I want to find the orthonormal paths $\psi_n$ that can diagonalize the second variation of the action $S[q]$. How to do that? Thanks
11
votes
7answers
1k views

Is physics rigorous in the mathematical sense?

I am a student studying Mathematics with no prior knowledge of Physics whatsoever except for very simple equations. I would like to ask, due to my experience with Mathematics: Is there a set of ...
2
votes
2answers
94 views

Constructing SUSY algebra via index structure

Often in literature the SUSY algebra is simply given, but various books, for example Bailin and Love, goes through the trouble of showing how the SUSY commutation relations are the only possible ones ...
2
votes
3answers
166 views

Configuration manifolds and constraints

In Classical Mechanics there's this notion of configuration manifold. Although I've heard about that a lot and although I often use that concept, I'm not sure I really understand them well because ...
11
votes
4answers
615 views

Can physics get rid of the continuum?

Almost every physical equation I can think of (even though I don't actually feel comfortable beyond the scope of classical mechanics and macroscopic thermodynamics, as that's enough for dealing with ...
6
votes
1answer
70 views

Are there any systems in physics which can only be formulated as an integral equation?

My question is are there any systems in physics that can only be formulated as an integral equation? Or do all integral equations have an equivalent differential equation?
0
votes
3answers
107 views

How would one begin to try to identify the properties of the mathematical structure corresponding to physical reality?

I was thinking about the Mathematical universe hypothesis and a natural question popped into my mind: Assuming that the universe (by universe I mean the complete physical reality) is really ...
6
votes
2answers
312 views

(Un)countability in QFT

I am a mathematician self-studying physics, and a currently working on QFT with Srednicki's book. One thing that bothers me is that for a scalar field (in the Hamiltonian version) there is a ...
2
votes
1answer
58 views

Mellin-Barnes (MB) integrals and hypergeometic functions

I'm trying to understand a step in arXiv:1104.2661. Equation 3.4 reads, \begin{equation} ...
3
votes
3answers
375 views

The Dirac-Delta function as an initial state for the quantum free particle

I want to ask if it is reasonable that I use the Dirac-Delta function as an intial state ($\Psi (x,0) $) for the free particle wavefunction and interpret it such that I say that the particle is ...
1
vote
2answers
835 views

Inverted Harmonic oscillator

what are the energies of the inverted Harmonic oscillator? $$ H=p^{2}-\omega^{2}x^{2} $$ since the eigenfunctions of this operator do not belong to any $ L^{2}(R)$ space I believe that the spectrum ...