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Questions tagged [mathematical-physics]

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 mathematical methods to problems in physics. Examples include partial differential equations (PDEs), variational calculus, functional analysis, and potential theory.

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18
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2answers
304 views

Can symmetry generators be used for quantization?

Take the Poincaré group for example. The conservation of rest-mass $m_0$ is generated by the invariance with respect to $p^2 = -\partial_\mu\partial^\mu$. Now if one simply claims The state where ...
102
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6answers
5k 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 ...
11
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2answers
757 views

Transforming a sum into an integral

I posted this in the mathematical forums. Maybe you will help me. I found an hard article http://prola.aps.org/abstract/PR/v105/i3/p776_1 of yang huang and luttinger. The authors begins with the sum: $...
6
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4answers
1k views

Generalized functions in physics

Prior to the Dirac delta function, what other distributions functions where physicists using? I find it hard to motivate the theory of generalized functions with just the delta function alone.
4
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0answers
217 views

K3 gravitational instanton

Could you please recommend a sufficiently elementary introduction to K3 gravitational instanton in general relativity and the problem of finding its explicit form? Under 'sufficiently elementary' I ...
4
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1answer
1k views

Orientability of spacetime

In many theoretical setups it is implicitly assumed that the underlying manifold (i.e. spacetime) is orientable. Then our analysis depends on this implicit assumption. For example, Stokes' theorem ...
6
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2answers
809 views

Positive Mass Theorem and Geodesic Deviation

This is a thought I had a while ago, and I was wondering if it was satisfactory as a physicist's proof of the positive mass theorem. The positive mass theorem was proven by Schoen and Yau using ...
5
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2answers
1k views

Normalizing the free particle wave function

One way to normalize the free particle wave function "is to replace the the boundary condition $\psi(\pm{\frac{a}{2}}) = 0$ [for the infinite well] by periodic boundary conditions expressed in the ...
5
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1answer
1k views

Are There Strings that aren't Chew-ish?

String theory is made from Chew-ish strings, strings which follow Geoffrey Chew's S-matrix principle. These strings have the property that all their scattering is via string exchange, so that the ...
4
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2answers
338 views

Combinatorial sum in a problem with a Fermi gas

I'm solving a problem involving a Fermi gas. There is a specific sum I cannot figure my way around. A set of equidistant levels, indexed by $m=0,1,2 \ldots$, is populated by spinless fermions with ...
3
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2answers
2k views

Does the phase space (configuration and momentum space) of particles have a Euclidean norm? Does it have a useful meaning of “distance”?

Often in engineering physics, different vector spaces are used to visualize the trajectories (evolution) of systems. An example being the 6n dimensional phase space of n particles. It is not very ...
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7answers
3k views

In quantum mechanics, given certain energy spectrum can one generate the corresponding potential?

A typical problem in quantum mechanics is to calculate the spectrum that corresponds to a given potential. Is there a one to one correspondence between the potential and its spectrum? If the answer ...
13
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5answers
617 views

Why must the field equations be differential?

In Landau–Lifshitz's Course of Theoretical Physics, Vol. 2 (‘Classical Fields Theory’), Ch. IV, § 27, there is an explanation why the field equations should be linear differential equations. It goes ...
3
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1answer
1k views

Numerical computation of the Rayleigh-Lamb curves

The Rayleigh-Lamb equations: $$\frac{\tan (pd)}{\tan (qd)}=-\left[\frac{4k^2pq}{\left(k^2-q^2\right)^2}\right]^{\pm 1}$$ (two equations, one with the +1 exponent and the other with the -1 exponent) ...
5
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2answers
272 views

differential equation with periodic conditions

given the differential operator Eigenvalue problem $ -D^{2}y(x)=E_{n}y(x) $ with boundary periodic conditions $ y(x)=y(x+1)$ my question is if there is a similar probelm for the Dilation operator $ ...
14
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1answer
667 views

How come random matrices can predict energy spectra of heavy atoms?

Some of the applications of random matrices is to find the spectra of heavy atoms in nuclear physics which are usually difficult to find otherwise. How can starting from randomness of some kind, ...
14
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6answers
1k views

Examples where an ill-behaved function leads to surprising results?

In mathematical derivations of physical identities, it is often more or less implicitly assumed that functions are well behaved. One example are the Maxwell identities in thermodynamics which assume ...
14
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1answer
269 views

Theorems on instability of classical systems of charged particles?

Classically, a hydrogen atom should not be stable, since it should radiate away all its energy. I remember hearing from my favorite freshman physics prof ca. 1983 about a general theorem to the effect ...
3
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2answers
633 views

Pade Approximant

I have some questions about Pade approximants. Given a divergent power series $ \sum_{n >0} a(n)x^{n} $ can we use a Pade Approximant to it $ R(x)$ so we can obtain a SUM of the series for every $...
11
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1answer
1k views

Discrete gauge theories

I'm trying to understand a particular case of gauge theories, namely discrete spaces on which a group G can act transitively, with a gauge group H which is discrete as well. From what I've already ...
14
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1answer
384 views

Can Fermionic symmetries be fully integrated into geometric deformation complexes or symplectic reduction?

How should a geometer think about quotienting out by a Fermionic symmetry? Is this a formal concept? A strictly linear concept? A sheaf theoretic concept? How does symplectic reduction work with odd ...
10
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3answers
2k views

What is a basis for the Hilbert space of a 1-D scattering state?

Suppose I have a massive particle in non-relativistic quantum mechanics. Its wavefunction can be written in the position basis as $$\vert \Psi \rangle = \Psi_x(x,t)$$ or in the momentum basis as $$...
4
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1answer
1k views

Physical significane and context in which Dirac introduced the Dirac Delta function

I'd like to know the exact context in which Paul Dirac introduced the Dirac delta function. What was the physical significance of the Dirac delta function when he first used it in Physics ?
5
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0answers
120 views

What can quantum adiabatic computation provably accomplish?

Let's say I have a quantum adiabatic computer in a black-box that works perfectly, doesn't suffer from decoherence/noise problems, etc. Are there any proven bounds for an adiabatic algorithm that ...
1
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0answers
175 views

Question on energies obtained via WKB approximation

Suppose we are given an ODE problem $$ y''(x)+V(x)f(x)=E_{n} y(x) $$ with boundary conditions $ y(0)=y(\infty)=0$. Here $V(x)$ is a potential function. Then is it always true that (for $n \...
14
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2answers
598 views

The entropic cost of tying knots in polymers

Imagine I take a polymer like polyethylene, of length $L$ with some number of Kuhn lengths $N$, and I tie into into a trefoil knot. What is the difference in entropy between this knotted polymer and ...
5
votes
2answers
482 views

Proof that Statistical Mechanics is a model of Themodynamics

The laws of thermodynamics are essentially four axioms of a mathematical theory. The expectation values of a statistical ensemble are supposed to satisfy the axioms of thermodynamics (under the ...
8
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1answer
629 views

The role of metric in the Wave Equation

The wave equation is often written in the form $$(\partial^2_t-\Delta)u=0,$$ involving the Laplace-Beltrami operator $\Delta$. However, the Laplace-Beltrami operator $\Delta$ is defined only in the ...
7
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1answer
693 views

Boundary Conditions Invariant Under Conformal Transformations in Electrostatics?

in two dimensional electrostatics it is assumed that the whole physical system is translationally invariant in one direction. Here, the two-dimensional Laplace equation $$\Delta \phi(x,y) = \frac{\...
2
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1answer
232 views

Did classical applications of density functional theory precede its use as an electronic structure method?

Density Functional Theory (DFT) is usually considered an electronic structure method, however a paper by Argaman and Makov highlights the applicability of the DFT formalism to classical systems, such ...
16
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3answers
2k views

What are the mathematical problems in introducing Spin 3/2 fermions?

Can the physics complications of introducing spin 3/2 Rarita-Schwinger matter be put in geometric (or other) terms readily accessible to a mathematician?
7
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1answer
851 views

Why Zeta regularization is not valid for multiple-loops?

Why zeta regularization only valid at one-loop? I mean there are zeta regularizations for multiple zeta sums. Also we could use the zeta regularization iteratively on each variable to obtain finite ...
2
votes
2answers
411 views

What is sector decomposition

What is sector decomposition and how can it be used to 'disentangle' UV and IR divergences? I have read about it in the paper SecDec: A general program for sector decomposition, but I have no idea if,...
2
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2answers
1k views

Operator relation involving the logarithm of an operator?

Dirac gives the relation: $\exp(iaq)f(q,p) = f(q, p - a\hbar)\exp(iaq)$ where $\hbar$ is Planck's constant. Can anybody give me the corresponding relation when the $\exp$ function is a $\ln$?
13
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2answers
3k views

Are there books on Regularization and Renormalization in QFT at an Introductory level?

Are there books on Regularization and Renormalization, in the context of quantum field theory at an Introductory level? Could you suggest one? Added: I posted at math.SE the question Reference ...
4
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2answers
1k views

Why was the truncated icosahedron (i.e. soccer ball) geometry chosen for the implosive lenses in the “Fat Man” atomic bomb?

Quoting from Wolfram Mathworld: " It is the shape used in the construction of soccer balls, and it was also the configuration of the lenses used for focusing the explosive shock waves of the ...
7
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10answers
7k views

Addition of different physical quantities

We all know the "apples and oranges" rule which says that it's meaningless to add or subtract two different quantities like apples and oranges. But the same rule doesn't hold for the multiplication ...
14
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4answers
3k views

Gelfand-Yaglom theorem for functional determinants

What is the 'Gelfand-Yaglom' Theorem? I have heard that it is used to calculate Functional determinants by solving an initial value problem of the form $Hy(x)-zy(x)=0$ with $y(0)=0$ and $y'(0)=1$. ...
10
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1answer
824 views

Iterated dimensional regularization

Given a 2-loop divergent integral $\int F(q,p)\,\mathrm{d}p\mathrm{d}q$, can it be solved iteratively? I mean I integrate over $p$ keeping $q$ constant Then I integrate over $q$ In both iterated ...
6
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2answers
533 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 ...
18
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5answers
3k 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, $...
2
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2answers
97 views

In a gas of particles, how is the displacement vector related to the number density?

Suppose I have a gas of particles that is initially uniformly distributed so that the number density is $n_0$ (number of particles per unit volume), and then I displace the particles by the vector ...
2
votes
1answer
224 views

Church–Turing Thesis

Can the Church–Turing Thesis be proved assuming classical mechanics, how is the proof or disproof? Edited: I was looking for a proof of "everything computable by a device obeying CM is computable by ...
2
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3answers
171 views

Length of a curve in D dimensional euclidean space

In a book I am reading on special relativity, the infinitesimal line element is defined as $dl^2=\delta_{ij}dx^idx^j$ (Einstein summation convention) where $\delta_{ij}$ is the euclidean metric. Next, ...
2
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2answers
1k views

How do you find conserved quantities for linear second order ODEs?

I have a differential equation of the form $ \frac{d^2 y}{dt^2} + f(t) \frac{dy}{dt} + g(t) y = 0 $ where $f$ and $g$ are known functions of time. Is there a systematic (or otherwise) way of ...
2
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2answers
310 views

a question on Lagrange's equation when the time derivative of the generalized co-ordinates is constant

Consider a system whose generalized co-ordinates are $q_i$ and is under the constraints $\dot{q_i} = K_i \forall i = 1,2,3,...$ where $K_i$ are constants. I have a problem in writing the Lagrange's ...
2
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1answer
880 views

significance of maxima and minima of time varying kinetic energy of a system

Consider a system of particles where the kinetic energy of the system is varying with time. I'd like to know the significance (or meaning) of the time derivative of the kinetic energy being zero at a ...
4
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1answer
2k views

Rayleigh-Lamb dispersion curves

In an infinite plane elastic plate of thickness $d$, it is shown that the modes of oscillation corresponding to a fixed time-frequency $\omega$ have wave-numbers given by solutions of the Rayleigh-...
4
votes
2answers
1k views

Helmholtz decomposition in the plane

Prove or disprove the following proposition: For any smooth plane vector field $\mathbf{H}=\left(H_x,H_y\right)$, there exist scalar potentials $\phi$, $\psi$ such that $H_x=\frac{\partial \phi }{\...
9
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1answer
4k views

Uniqueness of Helmholtz decomposition?

Helmholtz theorem states that given a smooth vector field $\pmb{H}$, there are a scalar field $\phi$ and a vector field $\pmb{G}$ such that $$\pmb{H}=\pmb{\nabla} \phi +\pmb{\nabla} \times \pmb{G},$$ ...