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 ...
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3answers
359 views
What's the meaning of the general solution and the particular solution in differential equations?
Can anybody cast some physical insight into this? I've been studying differential equations on my own and don't understand how you can have a whole host of general solutions. It seems like a rather ...
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vote
1answer
70 views
Smooth trajectory on a smooth manifold
Physicists talk about a smooth trajectory of a particle on a smooth manifold and they label it as q(t) where q_1(t)....q_n(t) are component functions coming from the homeomorphism. I don't see how we ...
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1answer
349 views
Schwinger representation of operators for n-particle 2-mode symmetric states
A bosonic (i.e. permutation-symmetric) state of $n$ particles in $2$ modes can be written as a homogenous polynomial in the creation operators, that is
$$\left(c_0 \hat{a}^{\dagger n} + c_1 ...
5
votes
3answers
248 views
Could general relativity and gauge theories in principle be covered in one course?
It's always nice to point out the structural similarieties between (semi-)Riemannian geometry and gauge field theories alla Classical yang Mills theories. Nevertheless, I feel the relation between the ...
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1answer
202 views
Why can't the functional integral be derived in a mathematically rigorous way?
Why can't the functional integral be derived in a mathematically rigorous way? What are the obstacles that we have to overcome in order to achieve that goal?
4
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2answers
598 views
Principal value integral
I am reading A. Zee, QFT in a nutshell, and in appendix 1 he has:
Meanwhile the principal value integral is defined by:
$$\int dx\,{\cal P}{1\over x}f(x)~=~ \lim_{\epsilon \rightarrow 0} \int ...
5
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0answers
163 views
Physical interpretation: weighted eigenvalues of the Laplacian with a potential
I'm a mathematician with only the basic knowledge of Physics, so my question may be trivial: in this case, mercy me. :-)
Let $\Omega \subseteq \mathbb{R}^N$ be a domain and let $V,m:\Omega \to ...
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2answers
156 views
Group rings in Physics
In my institute a few mathematicians work on Group Ring. Since it has close connection with representation theory, I thought that there must be some interesting connections of it with Physics. However ...
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1answer
82 views
When does the “norm of quasi-eigenvectors” matter in calculations? For which physical results are these even used?
Which physical system in nonrelativistic quantum mechanics is actually described by a model, where the norm of the "position eigenstate" (i.e. the delta distribution as limit of vectors in the ...
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0answers
120 views
How is the poincare conjecture(and perelman proof) helpful in studying the properties of the universe?
Can someone tell me how the poincare's famous conjecture or its proof by perelmen can be helpful in deciding some properties like the shape of the universe?
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3answers
437 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:
$$\frac{\delta F[f(x)]}{\delta f(y) } = \lim_{\epsilon \rightarrow 0} ...
3
votes
4answers
486 views
Topology needed for Differential Geometry [duplicate]
I am a physics undergrad, and need to study differential geometry ASAP to supplement my studies on solitons and instantons. How much topology do I need to know. I know some basic concepts reading from ...
2
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1answer
126 views
The (co)algebra for the (co)monad of a light switch
If we take a light switch to embody an entire category, we could take the light switch to be a set with two elements and the morphisms are all endofunctions. Let's say, for fun, that we define the ...
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1answer
505 views
Ideal gas with two kinds of particles, Grand canonical partition function
Consider an ideal gas contained in a volume V at temperature T. If all particles are identical the Grand canonical partition function can be calculated using
$$Z_g(V,T,z) := \sum_{N=0}^\infty z^N ...
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vote
1answer
168 views
Hairy ball theorem: references to applications
I'm looking for references to applications of the Hairy ball theorem.
This is a result of mathematics (topology), but I am interested in applications.
I already visited wikipedia and cited ...
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votes
2answers
374 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 ...
3
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1answer
191 views
interpretation of Green function
Is there a physical interpretation of the existence of poles for a Green function? In particular how can we interpret the fact that a pole is purely real or purely imaginary? It's a general question ...
5
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0answers
127 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 ...
6
votes
2answers
407 views
Introduction to string theory
I am in the last year of MSc. and would like to read string theory. I have the Zwiebach Book, but along with it what other advanced book can be followed, which can be a complimentary to Zwiebach. I ...
4
votes
1answer
121 views
Variational wavefunctions and “spread” of potential in quantum mechanics
A particle in a box has an energy that decreases with the size of the box. In the general case, it is often said that a variational solution for a "narrow and deep" potential is higher in energy than ...
2
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0answers
135 views
What are the topics of string theory that are comprehensible with only a mathematical background on Manifolds and Algebraic Topology?
What are the topics of string theory that are comprehensible with only a mathematical background on manifolds and algebraic topology? Also, I have read only the first four chapters in Peskin & ...
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vote
1answer
252 views
Wigner-Eckart projection theorem
I'm following the proof of Wigner-Eckart projection theorem which states that:
$$\langle \bf{A} \rangle ~=~ \frac{\langle \bf{A} \cdot \bf{J} \rangle}{\langle {\bf{J}}^2 \rangle} \langle \bf{J} ...
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2answers
641 views
EM wave function & photon wavefunction
According to this review
Photon wave function. Iwo Bialynicki-Birula. Progress in Optics 36 V (1996), pp. 245-294. arXiv:quant-ph/0508202,
a classical EM plane wavefunction is a wavefunction (in ...
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votes
2answers
378 views
Lie bracket for Lie algebra of $SO(n,m)$
How does one show that the bracket of elements in the Lie algebra of $SO(n,m)$ is given by
$$[J_{ab},J_{cd}]
~=~ i(\eta_{ad} J_{bc} + \eta_{bc} J_{ad} - \eta_{ac} J_{bd} - \eta_{bd}J_{ac}),$$
...
5
votes
1answer
277 views
Wigner-Eckart theorem of SU(3)
I have just come across the Wigner-Eckart theorem and am not sure on how to apply it. How do I find the matrix elements of $\langle u|T_a|v\rangle$ in terms of tensor components and the Gell-Mann ...
3
votes
1answer
227 views
How do I find the tensor components of all weights of a representation of SU(3), e.g. the six dimensional representation (2,0)
How do I find the corresponding tensor component v^ij of the six dimensional representation of SU(3) with dynkin label (2,0).
4
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1answer
997 views
Gabriele Veneziano, strong nuclear force and beta-function
Background to the question:
From The History of String Theory:
Gabriele Veneziano, a research fellow at CERN (a European particle accelerator lab) in 1968, observed a strange coincidence - many ...
4
votes
4answers
360 views
How to interpret the continuity conditions in the PDEs (for example, Maxwell equations) originated in physics?
I am currently working on PDEs in physics, mostly Maxwell equations. I am a mathematics graduate student, and this question has been haunting me for years.
In PDE theory, or more specifically the ...
17
votes
5answers
828 views
Is the converse of Noether's first theorem true: Every conservation law has a symmetry?
Noether's (first) theorem states that any differentiable symmetry of the action of a physical system has a corresponding conservation law.
Is the converse true: Any conservation law of a physical ...
9
votes
3answers
413 views
What areas of physics should a mathematician study to understand TQFT?
I am studying topological quantum field theory from the view point of mathematics.(axiomatic treatise) So it has no explanation about physics. I would like to know physic background of TQFT. But I ...
6
votes
1answer
113 views
precise definition of “moduli space”
I'm curious what the precise definition of the moduli space of a QFT is. One often talks about the classical moduli space, which then can get quantum corrections. Does this mean the quantum moduli ...
6
votes
2answers
93 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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0answers
260 views
Approximation of a summation by an integral
Is it valid to approximate the function $$ Z(t)=\sum_{n}e^{-tE_{n}} ,\ t\ge 0$$
by the integral over phase space: $$ \frac{ 1}{2\sqrt \pi}\int_{0}^{\infty}dxe^{-tV(x)}?$$
For example, in order to ...
7
votes
2answers
209 views
Advice on doing physics under the umbrella of mathematics and the converse
In the current scenario of research in QFT and string theory (and related mathematical topics), which of the following would an undergraduate student, like me, be advised to do and why if s/he is ...
2
votes
0answers
116 views
Functional determinant approximation
Let the Hamiltonian in one dimension be $H+z$, then I would like to evaluate $\det(H+z)$.
I have thought that if I know the function $Z(t) = \sum_{n>0}\exp(-tE_{n})$ I can use
$$\sum_{n} ...
5
votes
1answer
235 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 ...
18
votes
1answer
293 views
Geometric picture behind quantum expanders
A $(d,\lambda)$-quantum expander is a distribution $\nu$ over the unitary group $\mathcal{U}(d)$ with the property that: a) $|\mathrm{supp} \ \nu| =d$, b) $\Vert \mathbb{E}_{U \sim \nu} U \otimes ...
5
votes
2answers
601 views
Basic Question - Green's Functions in Quantum Mechanics
I am trying to learn about Green's functions as part of my graduate studies and have a rather basic question about them:
In my maths textbooks and a lot of places online, the basic Greens function G ...
8
votes
2answers
983 views
Proof that the One-Dimensional Simple Harmonic Oscillator is Non-Degenerate?
The standard treatment of the one-dimensional quantum simple harmonic oscillator (SHO) using the raising and lowering operators arrives at the countable basis of eigenstates $\{\vert n \rangle\}_{n = ...
2
votes
1answer
191 views
Equivalent system in Centre manifold theory
I was studying the centre manifold theory. It says (see Kuznetsov page 155, theorem 5.2) that the system on the left side of the picture is topologically equivalent to the one on the right.
$
...
7
votes
2answers
737 views
A book on quantum mechanics supported by the high-level mathematics
I'm interested in quantum mechanics book that uses high level mathematics (not only the usual functional analysis and the theory of generalised functions but the theory of pseudodifferential operators ...
2
votes
1answer
500 views
General procedure for Clebsch-Gordan expansions
I'm wondering if the Clebsch-Gordan series generalize to any orthonormal set of basis functions? If so, how would one go about deriving an expression for an arbitrary set of basis functions (perhaps ...
5
votes
4answers
755 views
Why can't any term which is added to the Lagrangian be written as a total derivative (or divergence)?
All right, I know there must be an elementary proof of this, but I am not sure why I never came across it before.
Adding a total time derivative to the Lagrangian (or a 4D divergence of some 4 ...
6
votes
4answers
453 views
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 ...
4
votes
1answer
36 views
Tip of a spreading wave-packet: asymptotics beyond all orders of a saddle point expansion
This is a technical question coming from mapping of an unrelated problem onto dynamics of a non-relativistic massive particle in 1+1 dimensions. This issue is with asymptotics dominated by a term ...
1
vote
5answers
177 views
Normal distribution of x, xdot
I have some real measurements from a process and I happened to look at the mutual distribution of (x(t), xdot(t)). I found that they seem to follow 2d normal distribution around (mu, 0). See image, ...
1
vote
1answer
318 views
Bra space and adjoint vectors
If I'm not wrong, a bra, $ \langle \phi_n | $, can be thought as a linear functional that when applied to a ket vector, $| \phi_m \rangle$, returns a complex number; that is, the inner product it's a ...
3
votes
2answers
219 views
How are functional determinants of Laplace-type operators used in physics?
Many mathematical papers concerning the $\zeta$-regularized Determinant of Laplace-type operators refer for motivation to the broad use of such determinants in mathematical physics, especially in ...
7
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3answers
375 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 ...
5
votes
3answers
1k views
What math do I need for mathematical physics? In what manner should I learn math? [closed]
I'm a freshman undergraduate. I've got my sight on mathematical physics. I love math but I don't have the talent nor the inclination for purely abstract mathematics. I also love physics.
The only ...
