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11
votes
3answers
448 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?
10
votes
4answers
427 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 ...
9
votes
1answer
648 views

The vacuum in quantum field theories: what is it?

In Section 10.1 of his textbook Quantum Field Theory for Mathematicians, Ticciati writes Assuming that the background field or classical source $j(x)$ is zero at space-time infinity, the presence ...
9
votes
1answer
1k views

“An operator is hermitian”. Implications?

Alastair Rae states that there are 4 postulates of Quantum Mechanics in his text on the subject matter. The first part of his second postulate can be stated as: Every dynamical variable may be ...
9
votes
9answers
1k views

Is causality a formalised concept in physics?

I have never seen a “causality operator” in physics. When people invoke the informal concept of causality aren’t they really talking about consistency (perhaps in a temporal context)? For example, if ...
8
votes
3answers
942 views

Use of 'complete' as in 'complete set of states' or 'complete basis'

Question. In the context of QM, I hear the phrases 'complete set of states' and 'complete basis' (among other similar expressions) thrown around rather a lot. What exactly is meant by 'complete'? ...
8
votes
2answers
613 views

Nonseparable Hilbert space

What kind of things can go wrong if we try to do quantum mechanics on a nonseparable Hilbert space? I have heard that usual mathematical manipulations that we take for granted will no longer hold. ...
7
votes
2answers
607 views

Vector Potential for Magnetic field when the field is not in simply-connected region

According to Poincare's Lemma, if $U\subset \mathbb{R}^n$ is a star-shaped set and if $\omega$ is a $k$-form defined in $U$ that is closed, then $\omega$ is exact, meaning that there's some ...
7
votes
3answers
1k views

Mathematical rigorous introduction to solid state physics

I am looking for a good mathematical rigorous introduction to solid state physics. The style and level for this solid state physics book should be comparable to Abraham Marsdens Foundations of ...
8
votes
2answers
246 views

Deriving the expectation of $[\hat X,\hat H]$

For a free particle of mass $m$, with Hamiltonian $$\hat{H} = \frac {\hat{P}^2} {2m},$$ where $$\hat{P} = -i \hbar \frac{\partial} {\partial x}.$$ The commutative relation is given by $$[\hat{X}, ...
6
votes
3answers
1k views

A question on the existence of Dirac points in graphene?

As we know, there are two distinct Dirac points for the free electrons in graphene. Which means that the energy spectrum of the 2$\times$2 Hermitian matrix $H(k_x,k_y)$ has two degenerate points $K$ ...
5
votes
3answers
295 views

Countable Matrix Representation

In my quantum mechanics class, my professor explained that the Hamiltonian along with position and momentum operators can be represented by matrices of countable dimension. This is especially usefull ...
4
votes
3answers
883 views

Comparison of 1D and 3D wave functions

When discussing the Schroedinger equation in spherical coordinates, it is standard practice in QM handbooks to point out that the radial part of the 3-dimensional wave equation bears a strong analogy ...
4
votes
5answers
667 views

If the size of universe doubled

My question is silly formulated, but I want to know if there is some sensible physical question buried in it: Suppose an exact copy of our universe is made, but where spatial distances and sizes are ...
2
votes
2answers
393 views

p-adic quantum mechanic [closed]

i got a degree on physics so my question is ?as a physicist could i learn P-adic analysis or p-adci quantum mechanics ?? is there any good book on the subject ? as an introductory level How the ...
0
votes
3answers
357 views

Given the Wikipedia notion of “arc length”, how is its manifestly real “signed variant” to be called and denoted?

I am dissatisfied with the presentation (not to say "definition") of "arc length", in its "Generalization to (pseudo-)Riemannian manifolds", as given in Wikipedia. (Who isn't?. But I'll sketch it here ...
12
votes
4answers
946 views

Rigorous proof of Bohr-Sommerfeld quantization

Bohr-Sommerfeld quantization provides an approximate recipe for recovering the spectrum of a quantum integrable system. Is there a mathematically rigorous explanation why this recipe works? In ...
7
votes
1answer
439 views

Spin-Statistics Theorem (SST)

Please can you help me understand the Spin-Statistics Theorem (SST)? How can I prove it from a QFT point of view? How rigorous one can get? Pauli's proof is in the case of non-interacting fields, how ...
4
votes
3answers
315 views

Guessing what a simple partial differential equation is describing physically

Is there an easy way to look at a partial different equation and get a sense of what kind of phenomena it is physically describing? I have an equation that looks like this: ...
1
vote
2answers
146 views

Good Fiber Bundles and Differential Geometry references for Physicists

I'm a student of Physics and I have interest on the theory of Fiber Bundles because of the applications they have in Physics (gauge theory for example). What are good books to learn the theory of ...
-1
votes
1answer
131 views

interaction between mathematical structures [closed]

From a physicist's perspective there are several situations in which somehow arbitrary choices of mathematical structures can be made. One can describe a system from different perspectives, etc. ...
8
votes
3answers
805 views

Boundary layer theory in fluids learning resources

I'm trying to understand boundary layer theory in fluids. All I've found are dimensional arguments, order of magnitude arguments, etc... What I'm looking for is more mathematically sound arguments. ...
6
votes
5answers
1k views

Hydrogen radial wave function infinity at $r=0$

When trying to solve the Schrödinger equation for hydrogen, one usually splits up the wave function into two parts: $$\psi(r,\phi,\theta)= R(r)Y_{l,m}(\phi,\theta).$$ I understand that the radial ...
6
votes
2answers
398 views

Lagrangians combining terms with 1 and 2 derivatives

How are field theory Langrangians treated when some terms have 2 derivatives but others have only 1? Because the number of derivatives in a Lagrangian term is more easily even than odd, the ...
4
votes
1answer
142 views

What exactly is the meaning of weak formulations and what is its purpose?

What is the purpose behind weak formulation of PDEs? I have read in the book by Zienkiewicz and Taylor that a weak formulation is more "permissive" that the original problem in the sense that it ...
1
vote
2answers
150 views

Why is quantum mechancis is not content with symmetric operators, but wants self-adjoint operators?

A symmetric operator has only real eigenvalues and different eigenvectors corresponding to different eigenvalues are orthogonal. These are exactly what we want for a physical observable. I think ...
77
votes
6answers
3k views

What are the justifying foundations of statistical mechanics without appealing to the ergodic hypothesis?

This question was listed as one of the questions in the proposal (see here), and I didn't know the answer. I don't know the ethics on blatantly stealing such a question, so if it should be deleted or ...
31
votes
6answers
763 views

Motion described by $m \frac{\mathrm{d}^2 x}{\mathrm{d}t^2}=-k\frac{\mathrm{d}^{\frac12 }x}{\mathrm{d}t^{\frac12}}$

What kind of motion would a (preferably dimensionless for simplicity) body do if the force acted on it was proportional to the semi-derivative of displacement, i.e. $$m \frac{\mathrm{d}^2 ...
14
votes
3answers
559 views

about the Atiyah-Segal axioms on topological quantum field theory

Trying to go through the page on Topological quantum field theory - The original Atiyah-Segal axioms - "Let $\Lambda$ be a commutative ring with 1, Atiyah originally proposed the axioms of a ...
13
votes
4answers
2k views

Where is the Atiyah-Singer index theorem used in physics?

I'm trying to get motivated in learning the Atiyah-Singer index theorem. In most places I read about it, e.g. wikipedia, it is mentioned that the theorem is important in theoretical physics. So my ...
10
votes
1answer
834 views

Topological insulators: why K-theory classification rather than homotopy classification?

I am reading a 2009 paper by Kitaev on K-theory classification of topological insulators. In the 4th page, 1st paragraph in the section "Classification principles", he says, Continuous ...
18
votes
1answer
1k views

Why is there a deep mysterious relation between string theory and number theory, elliptic curves, $E_8$ and the Monster group?

Why is there a deep mysterious relation between string theory and number theory (Langlands program), elliptic curves, modular functions, the exceptional group $E_8$, and the Monster group as in ...
35
votes
2answers
619 views

Physical interpretation of different selfadjoint extensions

Given a symmetric (densely defined) operator in a Hilbert space, there might be quite a lot of selfadjoint extensions to it. This might be the case for a Schrödinger operator with a "bad" potential. ...
14
votes
5answers
2k views

What does a frame of reference mean in terms of manifolds?

Because of my mathematical background, I've been finding it hard to relate the physics-talk I've been reading, with mathematical objects. In (say special) relativity, we have a Lorentzian manifold, ...
10
votes
1answer
803 views

Self-adjoint and unbounded operators in QM

An operator $A$ is said to be self-adjoint if $(\chi,A\psi)=(A\chi,\psi)$ for $\psi, \chi \in D_A$ and $D_A=D_{A^\dagger}$. But for the free particle momentum operator $\hat{p}$ these inner products ...
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 ...
13
votes
7answers
1k views

What are the uses of Hopf algebras in physics?

Hopf algebra is nice object full of structure (a bialgebra with an antipode). To get some idea what it looks like, group itself is a Hopf algebra, considered over a field with one element ;) usual ...
12
votes
2answers
373 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 ...
23
votes
3answers
2k views

Why are von Neumann Algebras important in quantum physics?

At the moment I am studying operator algebras from a mathematical point of view. Up to now I have read and heard of many remarks and side notes that von Neumann algebras ($W^*$ algebras) are important ...
19
votes
5answers
3k views

What do theoretical physicists need from computer scientists?

I recently co-authored a paper (not online yet unfortunately) with some chemists that essentially provided answers to the question, "What do chemists need from computer scientists?" This included the ...
15
votes
2answers
490 views

Generalized Complex Geometry and Theoretical Physics

I have been wondering about some of the different uses of Generalized Complex Geometry (GCG) in Physics. Without going into mathematical detail (see Gualtieri's thesis for reference), a Generalized ...
14
votes
2answers
808 views

Is the G2 Lie algebra useful for anything?

Seems like all the simpler Lie algebras have a use in one or another branch of theoretical physics. Even the exceptional E8 comes up in string theory. But G2? I've always wondered about that one. ...
9
votes
0answers
363 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+ ...
7
votes
1answer
322 views

Are identity types interpreted physically in an infinity-topos formulation of equations of motion?

In reference to Urs Schreibers paper/book on foundations of field theory Differential cohomology in a cohesive infinity-topos I wonder: are identity types there used "only" for the computations, or ...
6
votes
2answers
2k 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 ...
6
votes
1answer
713 views

CFTs and formalizing quantum field theory

Moshe's recent questions on formalizing quantum field theory and lattices as a definition of field theory remind me of something I occasionally idly wonder about, and maybe this site can tell me the ...
13
votes
0answers
350 views

What is Motivic mathematics and how is it used in physics?

In a few videos I've seen where he discusses the new approach to calculating the super Yang Mills scattering amplitudes, Nima Arkani-Hamed sometimes alludes to the use of Motivic methods as being ...
12
votes
1answer
276 views

Lie group of Schrodinger Wave equation

In Ballentine's book on quantum mechanics (in 3rd chapter), he introduces the symmetry transformation of Galilean group associated with Schrodinger equation. Now the Galilean group as such has 10 ...
10
votes
1answer
542 views

What's the physical intuition for symplectic structures?

I always thought about symplectic forms as elements of areas in little subspaces because of the Darboux theorem, however I cannot get the physical intuition for it and for the hamiltonian vector ...
8
votes
4answers
573 views

Using supersymmetry outside high energy/particle physics

Are there applications of supersymmetry in other branches of physics other than high energy/particle physics?