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What is a maximal analytic extension?

Can someone explain (as rigorously as possible) what is involved in analytically continuing, say, the Schwarzschild solution to the Kruskal manifold? I understand the two metrics separately but I'm ...
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3answers
414 views

How to think physically about basic “fields”

"Field" is a name for associating a value with each point in space. This value can be a scalar, vector or tensor etc. I read the wikipedia article and got that much, but then it goes it into more ...
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1answer
128 views

Intuition about Momentum Maps

I'm studying Classical Mechanics and there is one object that appeared recently on the book I'm not being able to get a physical intuition about it. The mathematical definition goes as follows: Let ...
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vote
1answer
200 views

Calculus of variations and string theory

In Polchinski's String theory book, Vol 1., in chapter 1, p. 18, he is deriving the Lagrangian in the light cone gauge (that's not necessary to know in order to answer this question), and he gets ...
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2answers
305 views

Does geodesic incompleteness in Penrose-Hawking theorems imply curvature blow up?

The singularity theorems in General Realtivity roughly stated say that given: A global causal condition An energy condition The existence of a closed trapped surface then spacetime must be ...
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votes
1answer
708 views

A confusion from Weinberg's QFT text (a vanishing term in Lippmann-Schwinger equation)

I was reviewing the first few chapters of Weinberg Vol I and found a hole in my understanding in page 112, where he tried to show in the asymptotic past $t=−∞$, the in states coincide with a free ...
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4answers
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Connection between Poisson Brackets and Symplectic Form

Jose and Saletan say the matrix elements of the Poisson Brackets (PB) in the $ {q,p} $ basis are the same as those of the inverse of the symplectic matrix $ \Omega^{-1} $, whereas the matrix elements ...
6
votes
1answer
312 views

Physical intuition/interpretation of fractional derivatives/integrals?

Oftentimes, when the derivative and integral operations are introduced within the realm of physics, we are taught some physical interpretation of them: Velocity is the derivative of position ...
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2answers
578 views

Can auxiliary fields be thought of as Lagrange multipliers?

In the BRST formalism of gauge theories, the Lautrup-Nakanishi field $B^a(x)$ appears as an auxiliary variable $$\mathcal{L}_\text{BRST}=-\frac{1}{4}F_{\mu\nu}^a F^{a\,\mu\nu}+\frac{1}{2}\xi B^a B^a + ...
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votes
2answers
252 views

fitting free QFTs into the Haag-Kastler algebraic formulation

Has the free Klein-Gordon quantum field theory been fitted into the Haag-Kastler algebraic framework? (Actually, John Baez told me "yes", and he should know.) If so, can you describe the basic ...
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2answers
753 views

What is the symmetry that corresponds to conservation of position?

We know that conserved quantities are associated with certain symmetries. For example conservation of momentum is associated with translational invariance, and conservation of angular momentum is ...
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1answer
155 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 ...
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votes
1answer
154 views

The index of a Dirac operator and its physical meaning

I recently read Witten's paper from the 1980s and he often uses the notion of the index of a Dirac operator in K-theory. What is the meaning of the index of a Dirac operator? What exactly is the ...
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0answers
184 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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votes
2answers
259 views

Why there is no Gibb's phenomenon in QM?

Why we don't see any Gibb's phenomenon in quantum mechanics? EDIT At sharp edges (discontinuities), we usually find ringing. This can be observed in many physical phenomenon (eg. shock waves). ...
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votes
1answer
228 views

Rigorous mathematical formalism of particle physics

Can anyone provide me with a rigorous mathematical definition of the fundamental particles (all fundamental bosons and fermions), reflecting the analogy of action of groups with interaction of ...
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0answers
119 views

What new does geometric or deformation quantization give to physics? [closed]

What new does geometric quantization or deformation quantization give to physics? For example: prediction of new physical phenomena or just better tool for quantization. What can these schemes do in ...
0
votes
1answer
291 views

Applying theorem of residues to a fermionic reservoir correlation function in order to solve the integral in the CF and obtain a summation

Applying theorem of residues to a fermionic reservoir correlation function in order to solve the integral in the correlation function and obtain a summation.
22
votes
2answers
149 views

Bell polytopes with nontrivial symmetries

Take $N$ parties, each of which receives an input $s_i \in {1, \dots, m_i}$ and produces an output $r_i \in {1, \dots, v_i}$, possibly in a nondeterministic manner. We are interested in joint ...
8
votes
2answers
326 views

Ambiguity in number of basis vectors [duplicate]

The dimension of the Hilbert space is determined by the number of independent basis vectors. There is a infinite discrete energy eigenbasis $\{|n\rangle\}$ in the problem of particle in a box which ...
8
votes
2answers
613 views

Is the step of analytic continuation unavoidable or can you model around it?

One sometimes considers the analytic continuation of certain quantities in physics and take them seriously. More so than the direct or actual values, actually. For example if you use the procedure ...
7
votes
1answer
286 views

Continuity domain for momentum operator

I know this is essentially a mathematic question, but I received no answer on math SE. Moreover it has a direct application in physics, so I thought to ask this here too. The momentum operator in one ...
7
votes
1answer
483 views

Thermodynamic limit “vs” the method of steepest descent

Let me use this lecture note as the reference. I would like to know how in the above the expression (14) was obtained from expression (12). In some sense it makes intuitive sense but I would ...
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0answers
110 views

Canonical second quantization vs canonical quantization with multisymplectic form in AQFT

First of all, I'm a mathematician that knows less than the basics of QFT, so forgive me if this question is trivial. Please, keep in my mind that my background in physics is very poor. 1) The usual ...
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votes
2answers
194 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. ...
6
votes
1answer
174 views

Are group representations possible when the solution space is not a vector space?

As far as I understand, the motivation for using representation theory in high energy physics is as follows. Assume that a theory has some (internal or external) symmetry group which acts on a vector ...
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0answers
297 views

Prequisites to learn Topological Field Theory? [closed]

Sorry for the somewhat qualitative question but what are the essential prerequisites for someone wanting to learn topological field theory from say the more physical side of things? The math side also ...
6
votes
1answer
2k views

What are Wightman fields/functions

Simple question: What are Wightman fields? What are Wightman functions? What are their uses? For example can I use them in operator product expansions? How about in scattering theory?
6
votes
1answer
372 views

Grassmann Variables Representation?

It might be a silly question, but I was never mathematically introduced to the topic. Is there a representation for Grassmann Variables using real field. For example, gamma matrices have a ...
6
votes
1answer
598 views

There seems to be no definition of “stability” in axiomatic QFT. Is there? And, if not, is this a problem?

"stability" is invoked as the justification for the axiomatic requirement that the spectrum of the generators of the translation group must be confined to the forward light-cone. The spectrum ...
5
votes
1answer
160 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 ...
5
votes
0answers
228 views

explicit matrix elements for a representation decomposed into subgroup by branching rules

I'm looking for a way to construct a representation for a simple Lie group such that one particular subgroup is manifest. I learned the branching rules from Cahn, Georgi and Slansky, but I'm still not ...
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votes
2answers
1k views

Jauch, Piron, Ludwig -> QFT? [duplicate]

Possible Duplicate: What is a complete book for quantum field theory? At the moment I am studying Piron: Foundations of Quantum Physics, Jauch: Foundations of Quantum Mechanics, and ...
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3answers
1k views

Books for linear operator and spectral theory

I need some books to learn the basis of linear operator theory and the spectral theory with, if it's possible, physics application to quantum mechanics. Can somebody help me?
4
votes
2answers
111 views

Amplitude-phase decomposition as a canonical transformation

I am studying a classical dynamical system defined on $\mathbb{CP}^2$: the phase space is parametrized in terms of three complex coordinates $\psi_i$ ($i=1,2,3$) and Hamilton's equations of motion ...
4
votes
0answers
233 views

Is the Hilbert space spanned by both bound and continuous hydrogen atom eigenfunctions?

As e.g. Griffiths says (p. 103, Introduction to Quantum Mechanics, 2nd ed.), if a spectrum of a linear operator is continuous, the eigenfunctions are not normalizable, therefore it has no ...
3
votes
1answer
98 views

Is there a general theorem stating why the restricted Lorentz group's exponential map is surjective?

The exponential map for the restricted Lorentz group is surjective. An outline of why is shown on the wiki page Representation Theory of the Lorentz Group. Is there a more general theorem that states ...
3
votes
1answer
380 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 ...
3
votes
3answers
519 views

Physical significance of getting an non-integrable function in an equation

I just found out during my Calculus course in High School, that there exist functions which cannot be integrated. Then I thought that I come across a lot of integrals while solving Physics ...
3
votes
4answers
812 views

Why the Hamiltonian and the Lagrangian are used interchangeably in QFT perturbation calculations

Whenever one needs to calculate correlation functions in QFT using perturbations one encounters the following expression: $\langle 0| some\ operators \times \exp(iS_{(t)}) |0\rangle$ where, ...
2
votes
2answers
205 views

If path integrals aren't well-defined, how can they have any physical meaning?

I am confused about a particular point about the nature of path integration. According to what I've read, what we really mean when we say functional integration is \begin{equation} ...
2
votes
0answers
106 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 ...
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vote
0answers
29 views

Is there a Bayesian theory of deterministic signal? Prequel and motivation for my previous question

This is a prequel to my question: What's the probability distribution of a deterministic signal or how to marginalize a dynamical system? Clearly my question looks at the same time fairly ...
1
vote
1answer
167 views

Continuous spectrum of hydrogen atom

I wonder if there is a nice treatment of the continuous spectrum of hydrogen atom in the physics literature--showing how the spectrum decomposition looks and how to derive it.
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1answer
682 views

State normalization in Dirac's formulation of quantum mechanics

Let us divide the time $T$ into $N$ segments each lasting $δt = T/N$. Then we write $\langle q_F | e^{−iHT} |q_I \rangle = \langle q_F | e^{−iHδt} e^{−iHδt} . . . e^{−iHδt} |q_I \rangle $ Our ...
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2answers
194 views

How much of the Capelli-Itzykson-Zuber ADE-classification of su(2)-conformal field theories can one see perturbatively?

In their celebrated work, Capelli Itzykson and Zuber established an ADE-classification of modular invariant CFTs with chiral algebra $\mathfrak{su}(2)_k$. How much of that classification can one ...
11
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2answers
219 views

Discussions of the axioms of AQFT

The most recent discussion of what axioms one might drop from the Wightman axioms to allow the construction of realistic models that I'm aware of is Streater, Rep. Prog. Phys. 1975 38 771-846, ...
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2answers
2k views

How important is mathematical proof in physics?

How important are proofs in physics? If something is mathematically proven to follow from something we know is true, does it still require experimental verification? Are there examples of things that ...
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2answers
802 views

Interesting topics to research in mathematical physics for undergraduates

I'm planning on getting into research in mathematical physics and was wondering about interesting topics I can get into and possibly make some progress on. I'm particularity fond of abstract algebra ...
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5answers
735 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 ...