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2answers
91 views

System without ground state is not real in nature?

We know that Coulomb force is real phenomena in nature and with Coulomb potential $V(x) \thicksim -\frac{1}{|x|}$ lowest energy is bounded in hydrogen atom. But it's mathematically clear that if ...
3
votes
0answers
70 views

Density matrix formalism and group representation

The postulates of quantum theory can be given in the density matrix formalism. States correspond to positive trace class operators with trace 1 on a Hilbert space $\mathcal{H}$. Composition is defined ...
3
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1answer
57 views

Relation between solutions to Yang-Baxter equations, integrability and exact solvability?

Wikipedia mentions that there is an implication: Yang-Baxter solutions yield integrable models, what 1D systems concerns. In arbitrary dimensions, what is the relation, if any, between solutions to ...
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3answers
515 views

What is the Hilbert space of a single electron?

Is it the same as the space of all possible descriptions of a single electron? If not, how do they differ? Please give the mathematical name or specification of this space or these spaces.
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0answers
45 views

How to carry out the perturbation expansion of an anharmonic oscillator to high orders?

I think this is a standard problem in quantum mechanics. Consider the anharmonic oscillator $E \psi = \left(- \frac{1}{2} \frac{\partial^2}{\partial^2 x } + \frac{1}{2}x^2 + \epsilon x^4 \right) ...
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0answers
558 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 ...
3
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3answers
572 views

Intuition for Path Integrals and How to Evaluate Them

I'm just starting to come across path integrals in quantum field theory, and want to get the right intuition for the them from the start. The amplitude for propagation from $x_a$ to $x_b$ is typically ...
5
votes
3answers
448 views

Curvature of Conical spacetime

Inspired by: Angular deficit The 2+1 spacetime is easier for me to visualize, so let's use that here. (so I guess the cosmic string is now just a 'point' in space, but a 'line' in spacetime) Edward ...
4
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0answers
128 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 ...
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0answers
21 views

Can isotropy (or anisotropy) be expressed in terms of intervals ($s^2$) between pairs of events?

Considering a set $\mathcal S$ of events and given the values of intervals $s^2[~P, Q~] \in \mathbb R$ for all pairs of events $P, Q \in \mathcal S$ (up to a common non-zero scale factor): how can ...
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1answer
32 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 ...
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3answers
284 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( ...
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3answers
75 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
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1answer
120 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) = ...
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2answers
168 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
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1answer
66 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
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1answer
171 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
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1answer
77 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
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1answer
88 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 ...
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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
631 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
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0answers
62 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
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1answer
74 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 ...
5
votes
2answers
140 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
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0answers
41 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 ...
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1answer
45 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
1answer
63 views

Relationship between Connection and Material Derivative

Suppose $D\subset \Bbb R^3$ contains a fluid and that $f : D\times \mathbb{R}\to \mathbb{R}$ is a time dependent function defined on the fluid region. In that case, the material derivative is defined ...
2
votes
2answers
119 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
180 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
119 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
649 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
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1answer
68 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 ...
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2answers
129 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
902 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
628 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 ...
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votes
2answers
413 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
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1answer
73 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
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2answers
355 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
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0answers
58 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
108 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
155 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
59 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
109 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
820 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
44 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 ...
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3answers
346 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
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1answer
186 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
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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
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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
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0answers
130 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 ...