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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4
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2answers
421 views

Confusion with Virtual Displacement

I have just been introduced to the notion of virtual displacement and I am quite confused. My professor simply defined a virtual displacement as an infinitesimal displacement that occurs ...
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0answers
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How to do the classification of Yangian invariant?

From Nima's paper:Scattering Amplitudes and the Positive Grassmannian arxiv:1212.5605 page91, we can see that there is a complete classification for $k=2$ Yangian inariants. But I have two questions ...
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1answer
276 views

Commutator of rotation matrices

How do you compute the commutator of rotation matrices in two different directions by different angles? Let $R_{x}(\alpha)$ be the rotation matrix about the $x$-axis and $R_{z}(\beta)$ be the ...
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Dervie a lagrangian formulation from an eulerian One

in the solidification of a binary alloy : X18%Y . where the nominal concentration of Y in the alloy phase is 18% , the goal is to determine the variation of < C> ;the average concentration over ...
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2answers
104 views

Why isn't the Segal-Bargmann space used more often in Quantum Mechanics?

The Stone-von Neumann Theorem states that a Hilbert space on which is defined an irreducible set of operators which satisfy the exponentiated canonical commutation relations is unitarily equivalent to ...
4
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1answer
225 views

Do translation formulae for generalised solid spherical harmonics exist?

I'm aware of the solid spherical harmonics functions, which are basically the surface spherical harmonics $Y^m_{\ell}(\theta,\varphi)$ with an additional monomial term along the radial direction: $R^...
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6answers
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Haag's theorem and practical QFT computations

There exists this famous Haag's theorem which basically states that the interaction picture in QFT cannot exist. Yet, everyone uses it to calculate almost everything in QFT and it works beautifully. ...
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51 views

Order parameter fluctuations in the mean field model for ferromagnetism (mathematical approach)

I'm a math student taking first steps into statistical mechanics and... I need help! Consider the Curie-Weiss model (i.e. the classical mean field model for ferromagnetism). If $N$ is the number of ...
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0answers
52 views

Fidelity of unitary operators: is $||U-\tilde U|| < \delta$ a *necessary* and *sufficient* condition?

There's a notion of fidelity of quantum states. However, is there a standard notion of the fidelity of unitary operators? Say, I wish to approximate a unitary operator $U$ acting on $n$ qubits with a ...
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4answers
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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 $y'(0)=1$. ...
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6answers
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Reason for the discreteness arising in quantum mechanics?

What is the most essential reason that actually leads to the quantization. I am reading the book on quantum mechanics by Griffiths. The quanta in the infinite potential well for e.g. arise due to the ...
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0answers
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Is there a useful relationship between connection on space coordinates and material derivative?

I am referring to an important part of the question Relationship between Connection and Material Derivative. Here is a paste and cut of the relevant part. That is the directional derivative along $\...
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1answer
101 views

Why do we need conformal compactification to define the global conformal group?

First I have the definition of a conformal map. Let $(M,g)$ and $(M',g')$ be two pseudo-Riemannian manifolds of same dimension. Let $U\subset M$ and $V\subset M'$, we say that a smooth map of maximal ...
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1answer
162 views

Why there's a Lorentz inner product in the unitary representations of the translation group?

Consider Minkowski spacetime. Its translation group is just the additive group $\mathbb{R}^4$. This is an abelian locally compact group. Next, consider one unitary representation $T : \mathbb{R}^4\to ...
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2answers
143 views

Does the convergence of van Hove sequence equivalent to thermodynamic limit?

I have seen two definitions of thermodynamic limit. Definition 1. This is the common definition in statistical physics textbooks, $$N \to \infty ,V \to \infty ,N/V = {\rm{constant}}$$ There are many ...
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42 views

Is the slater determinant the unique ground state of a system of independent $N$ electrons?

If $H = \sum_{i = 1}^N h_i$ is the hamiltonian for a system of $N$ independent electrons, is the ground state for this system unique? My tutor seems to think so because he writes «The ground state of ...
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77 views

How do I use calculus of variations to prove that Carnot cycle is the most efficient one?

There are already many questions on Physics SE asking for a proof of the maximum possible efficiency of Carnot's engine. However, none of them (at least the ones which I looked up) contain a ...
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1answer
178 views

What classifies gaugings?

Gauging a global symmetry $G$ introduces several free parameters to the theory. For example, In $d=4$, gauging a simple and simply-connected Lie group introduces a coupling constant and a theta term, ...
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1answer
42 views

How to find convergence of conditionally convergent series obtained while calculating the electrostatic potential energy of a NaCl crystal?

I was reading Electricity and Magnetism by E M Purcell and there in the first chapter there is an attempt to estimate the electrostatic potential energy of the crystal lattice of a NaCl crystal. ...
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1answer
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Why is it said that all the intrinsic information in a self-adjoint (or Hermitian) operator is contained in its spectrum?

Let $A$ be a self-adjoint operator. Then, we can define the pure point spectrum of $A$ $$\sigma_{\mathrm{pp}}(A) := \{z\in \Bbb C \ | \ \mathrm{ran(A-z)} = \overline{\mathrm{ran (A-z)}} \neq \...
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1answer
157 views

Why time-dependent canonical transformation satisfies symplectic condition?

I am reading Chapter 9 of Goldstein. He proves that any time-independent canonical transformations satisfy symplectic condition. And after that, he shows that if we ignore second order small quantity, ...
6
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1answer
100 views

Application of a non-matrix Lie group?

I'm studying Lie theory from Brian C. Hall's "Lie Groups, Lie Algebras, and Representations," in which he focuses on matrix Lie groups (defined as sets of matrices) rather than general Lie groups (...
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Advice needed on learning maths oneself [duplicate]

I am a master's student in physics trying to learn maths on my own. My classes workload is heavy and the schedule is very hectic due to which I don't get time to do mathematics. I have done Linear ...
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1answer
46 views

“…because there is no saturation.” Elliott Lieb on the interacting 1d Bose gas

In the this article by Lieb, Liniger: "Exact Analysis of an Interacting Bose Gas. I. The General Solution and the Ground State" a repulsive Bose gas is considered in 1d with Hamiltonian $$ H = -\sum_\...
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1answer
151 views

Why is the linear momentum a moment map?

Consider a particle with mass $m$ moving in a constant speed $v$ along the real line $\mathbb{R}$, with coordinate $q$. Then its linear momentum is $p = mv$. Now let's translate the physics into math,...
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What is the momentum mapping for a left translation group action?

This question is a follow-up to a previous question. In this survey of Symplectic Geometry by Arnol'd and Givental, the example of a constant-speed moving particle considered in the previous question ...
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2answers
80 views

Why in an irreducible unitary representation of the Poincare group all momenta are on the same mass shell?

This is a question about the approach of Weinberg in "The Quantum Theory of Fields" to the irreducible unitary representations of the Poincare group in Chapter 2. Let $U(\Lambda,a)$ be such a ...
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0answers
25 views

How would one solve this system of differential equations for muzzle velocity as a function of initial pressure?

I'm trying to solve for the exit velocity of a projectile of mass m from a t-shirt cannon with pressurized air tank with initial pressure $P_0$ but I am stuck on how to solve either analytically or ...
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0answers
44 views

KMS condition and quasi-free states

In algebraic formulation of QFT, it is known that if a state is KMS with respect to some time parameter $\tau$, then the Wightman 2-point functions must satisfy certain conditions, namely stationarity ...
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0answers
77 views

Witten's description of WZW conformal blocks

I am reading this paper by Witten - Geometric Langlands From Six Dimensions. In section 4.1, he gives a description of the vector space of conformal blocks of the current algebra associated to a ...
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1answer
57 views

A driver approaches a traffic light [closed]

A driver approaches a traffic light, which is green with speed $ v_0 $ when it turns yellow. a) If the driver's reaction occurs within temp $ T $, during which he decides to stop and apply the brake ...
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122 views

Where do theta terms live?

Consider a gauge theory with group $G$. The canonical kinetic term for the gauge field is $F\wedge\star F$ and, depending on the dimensionality of spacetime, there are other allowed terms, such as ...
3
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1answer
58 views

The AQFT defined by ordinary NRQM

It is often said that NRQM is one dimensional QFT. The Haag-Kastler axioms for QFT should apply to NRQM also then.* So, for the NRQM system $L^2\left(\mathbb{R}^3\right)$ with time evolution given ...
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Groups with cardinality larger than the Reals in physics

In what physical theories are sets with cardinality larger than $\aleph_1$ used? There are plenty of examples of finite, countable, and uncountable vector spaces in physics, but do physicists ever ...
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1answer
59 views

Which are the best mathematical methods books for which topic for a physics undergrad? [duplicate]

I am a physics undergraduate and I would be glad if you share your opinion about which books are best for which topics in mathematical methods, from very basic to advanced. (Like you some say Tom ...
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7answers
10k views

Rigor in quantum field theory

Quantum field theory is a broad subject and has the reputation of using methods which are mathematically desiring. For example working with and subtracting infinities or the use of path integrals, ...
6
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1answer
320 views

Is there a function which is square integrable and doesn't tend to zero at infinity but it belongs in the domain of the momentum operator?

Is there a function which is square integrable and doesn't tend to zero at infinity but it belongs in the domain of the momentum operator? There are some counterexample for functions that are square-...
3
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1answer
2k views

What is the mathematical background needed for quantum physics? [duplicate]

I'm a computer scientist with a huge interest in mathematics. I have also recently started to develop some interest about quantum mechanics and quantum field theory. Assuming some knowledge in the ...
3
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0answers
60 views

Initial value problem on $\mathcal{I}^-$ for Maxwell fields

In the paper "Symplectic Geometry of Radiative Modes and Conserved Quantities at Null Infinity" by Ashtekar and Streubel the authors state the following: Fix, as in § 2(a), a conformal completion $(...
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2answers
71 views

Stone’s Theorem and Time Ordered Exponentials

The time evolution operator of quantum mechanics seems (at least to me) to form a strongly continuous, one parameter group of unitary operators. Hence, by Stone’s theorem, we should have that $U(t) = \...
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0answers
27 views

Topological term in the Lieb-Liniger model: two different excitations map to the same state?

The many body Lieb-Liniger Hamiltonian is defined by ($c>0$ throughout ) \begin{equation} \hat H = -\sum_\ell \partial_{\ell}^2 + 2c\sum_{\ell}\sum_{m<\ell} \delta(x_{m}-x_\ell). \end{equation} ...
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0answers
36 views

Growth of apparant horizons and null convergence condition

An apparent horizon in general relativity is a surface where all null vectors are pointing "inwards", i.e. it is the location of a marginally outer trapped surface (for a review see here). It is well ...
4
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0answers
248 views

Any real contribution of functional analysis to quantum theory as a branch of physics? [closed]

In the last paragraph of this last paper of Klaas Landsman, you can read: Finally, let me note that this was a winner's (or "whig") history, full of hero-worship: following in the footsteps of ...
84
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6answers
12k views

Trace of a commutator is zero - but what about the commutator of $x$ and $p$?

Operators can be cyclically interchanged inside a trace: $${\rm Tr} (AB)~=~{\rm Tr} (BA).$$ This means the trace of a commutator of any two operators is zero: $${\rm Tr} ([A,B])~=~0.$$ But what about ...
6
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2answers
192 views

Correspondence between integral transformations and differential operators in quantum mechanics

The translation operator in one dimension is defined as $$ \hat T \psi(x) = \psi(x-\alpha) . $$ This can be written as an integral transformation, $$ \begin{align*} \hat T\psi(x) = \langle x|\hat ...
4
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2answers
127 views

How does one “invert” such infinite-dimensional sympletic form?

In the book "Lectures on the IR structure of gravity and gauge theories" by Strominger the author considers the sympletic form for free electrodynamics: $$\Omega_\Sigma[A;\delta_1 A,\delta_2A]=-\frac{...
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4answers
2k views

Does Hestenes Zitterbewegung Explain why complex numbers appear in QM?

This question may fit better in the discussion of "Why Complex variables are required by QM?", but it also relates to the extent to which arguments by Hestenes are accepted in mainstream physics and ...
2
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1answer
89 views

Partial trace over continuous degrees of freedom

Let us suppose we have some quantum system whose Hilbert space admits a bipartition $\mathscr{H}\simeq \mathscr{H}_A\otimes \mathscr{H}_B$. Let $|n\rangle_A$ be a basis of $\mathscr{H}_A$ and $|m\...
10
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2answers
1k views

Solution for inverse square potential in $d=3$ spatial dimensions in quantum mechanics [duplicate]

Can a particle in an inverse square potential $$V(r)=-1/r^{2}$$ in $d=3$ spatial dimensions be solved exactly? Also please explain me the physical significance of this potential in comparison with ...
0
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0answers
53 views

The mathematical structure of $\widehat{su(2)}_k$

Some of my colleagues work on CFT's and quantum groups and I hear them talk a lot about $\widehat{su(2)}_k$ algebras. According to them (and the general physics literature) these are what ...

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