Questions tagged [mp.mathematical-physics]

Mathematical methods in classical mechanics, classical and quantum field theory, quantum mechanics, statistical mechanics, condensed matter, nuclear and atomic physics.

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14 views

Invertibility of discrete Laplacian

In QFT and Statistical Mechanics the discrete Laplacian usually plays a key role when we want to discretize the theory. However, few books (at least to my knowledge) really work the properties of this ...
4
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1answer
103 views

1d TQFT minus connection =?

Correct me if I am wrong but I believe at least conceptually (maybe even rigorously) data of a 1-dimensional TQFT and of a vector bundle with connection are equivalent. Going into more detail (and ...
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43 views

Characterizing Lagrangian submanifolds of odd-symplectic manifold

Theorem 4.57 & 4.62 of Mnev's paper BV formalism and applications state the following: Theorem 4.57 (ii) in Mnev's paper Let $(\mathcal M, \omega)$ be an odd-symplectic manifold with body $...
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86 views

Physics applications of quantum logic

Are there any examples of quantum logic being applied to solve actual physical questions, in particular to predict the physical properties (spectrum etc.) of some quantum-mechanical system? (Note that ...
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24 views

How to proceed in this Boundary value problem where Eigen values are calculated numerically?

While solving a boundary value problem (background provided in the Context section) I reach the following variable separated two equations ($F(x)$ and $G(y)$) \begin{eqnarray} \lambda_h F''' - 2 \...
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1answer
94 views

Anti-symmetric operators for the Dirac or Majorana spinors

In a Zoom lecture given by a mathematical physics professor, if I recalled correctly, he explained that the in 1+1 dimensional spacetime (or 2 dimensions in short), the "action" of fermions (spinors) ...
7
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2answers
199 views

Path integral derivation of extended TQFT

I know this isn't exactly a math question, but I am asking it here anyway. We define an extended TQFT to be a functor (preserving tensor products) from the $\left(\infty,n\right)$-category of ...
4
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144 views
+50

Batalin-Vilkovisky integral is invariant under infinitesimal deformation

This is from page 90 and 93 of Mnev's paper BV formalism and applications. Let $\mathcal L_{t} \subset \Pi T^{*}M$ be a smooth family of Lagrangians with $t \in [0,1]$ a parameter, s.t. $\...
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2answers
280 views

Exponential decay of voltage potential difference

Consider the following adjacency matrix of a complete graph: $$A=(e^{-|i-j|})_{1\leq i\neq j\leq n}$$ with 0 on the diagonal. Let $D=diag\{d_1,...,d_n\}$ be the degree matrix where $d_i=\sum_{j\neq i}...
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1answer
125 views

Question about Berezin line bundle of odd cotangent bundle of supermanifold $\text{Ber}(\Pi T^*\mathcal N)$

The followings are from Mnev's paper about BV formalism. Example 4.15 (Definition of split supermanifold) Let $E \to M$ be a rank $m$ vector bundle over $n$-manifold $M$, then there exists a ...
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1answer
99 views

Maximally symmetric hyperbolic 3-manifolds with finite volume

In physics, standard cosmology is build with simple maximally symmetric 3-manifolds (spacelike time-slices of constant curvature, e.g. $S^3$ or less popular the hyperbolic space $H^3$). Since $S^3$ ...
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1answer
100 views

Quotient distance on $\mathbb{C}P^n$ is equivalent to distance induced by the Fubini-Study metric

Let $n$ be an even, positive integer. Then, is the metric (in the metric-space sense) on $\mathbb{C}P^n$ induced by the Fubini-Study (Riemannian) metric equivalent to the quotient (pseudo?)-metric ...
15
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1answer
569 views

Current status of axiomatic quantum field theory research

Axiomatic quantum field theory (e.g. the wightman formalism and constructive quantum field theory) is an important subject. When I look into textbooks and papers, I mostly find that the basic ...
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0answers
88 views

Book on Rigorous Renormalization

Many years ago I came across Salmhofer's Renormalization book and I studied its first chapter for a while. At the time, a professor told me the aim of the book was to develop a perturbative fermionic ...
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1answer
137 views

Non-perturbative Renormalization in the sense of Polchinski's equation. Do we have a mathematical formulation?

My question is about mathematical treatment of exact renormalization group in the sense of Polchinski's flow equation. In a heuristic form, Polchinski's equation looks like: $\partial_t S[\phi] = \...
3
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1answer
105 views

More important or relevant progress in discretizing hard problems in physics in last decade

This is a reference request, and soft question as companion. I'm curious to ask, from an informative point of view, what about the more important progress in the goal to discretize hard problems in ...
4
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215 views

Inequalities for trace/eigenvalues of product of multiple 2x2 matrices

Consider the matrix product $\prod_i^n A_i$, where each $A_i$ is a $2\times2$ matrix having the form $A_i = \left( \begin{smallmatrix} \lambda + \alpha_i & -\beta_i \\ 1 & 0\end{smallmatrix}\...
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1answer
126 views

$\mathbb Z$-graded manifold is isomorphic to the structure sheaf of supermanifold locally

In this paper, definition 4.4.1 about supermanifold and definition 4.6.1 about graded manifold: Definition 4.4.1: An supermanifold $\mathcal{M}$ is a locally ringed space $(M,\mathcal O_M)$ which ...
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1answer
279 views

What is the relevant literature –if any– on real-valued functions on sets and their Boolean combinations? [closed]

As part of a project (https://arxiv.org/abs/2004.06745), I've constructed the following table, $\left( \begin{array}{ccc} \hline Constraint Imposed & Probability & Quasirandom Estimate \\ ...
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0answers
35 views

Complete list of indecomposable representations of Temperley-Lieb algebras at roots of unity?

The Temperley Lieb algebra $TL_n$ at roots of unity is not semisimple. The standard representations $V_{n,p}$ are indecomposable but, in general, not irreducible. If $K_{n,p}$ is the sub-...
4
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1answer
203 views

Mirzakhani's hyperbolic method generalized to moduli space of stable maps

I've been learning about Mirzakhani's use of hyperbolic geometry to compute Weil-Petersson volumes of moduli space of curves, and the application to proving Virasoro constraints for a point. Why have ...
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0answers
45 views

Approach to solve a coupled system of PDE (heat transfer in cylindrical coordinates)

I have the following two PDEs, which describe steady-state coupled heat transport between an externally heated axisymmetric solid body (Eq. 1, $T(r,z)$) and a fluid (Eq. 2, $t(z)$) flowing inside it: ...
5
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1answer
162 views

Jack function in power symmetric basis

In Macdonald's book, the Jack symmetric function $J_{\lambda}(x_1,\ldots, x_n)$ for a partition $\lambda$ is defined by three properties (orthogonality, triangularity, and normalization). In the ...
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2answers
456 views

Less fundamental applications of Zeta regularization:

As we all know, zeta regularization is used in Quantum field theory and calculations regarding the Casimir effect. Are there less fundamental applications of zeta function regularization? By "less ...
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0answers
77 views

Bifurcation points in parametric Hammerstein Integral equation

I am a theoretical physicist working on integrable systems, hence I preemptively ask forgiveness for the eventual lack of mathematical rigor. My question concerns the properties of a particular ...
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0answers
38 views

Interpret certain expressions in terms of classical quadratic surfaces

I have two primary constraints, a linear one, \begin{equation} \label{C1} C_1=Q_1>0\land Q_2>0\land Q_3>0\land Q_1+3 Q_2+2 Q_3<1, \end{equation} and a quadratic one (incorporating $C_1$), \...
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0answers
78 views

Geometrical proof of Noether Theorem [duplicate]

I am reading a very nice Physics book "The standard model in a nutshell" by D.Goldberg and just read there a mention to Noether Theorem. Of course I knew this outstanding theorem very well from ...
4
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1answer
85 views

Effective action, partition function and the renormalization group

Mayer expansions and the Hamilton–Jacobi equation by D. Brydges and T. Kennedy begins mentioning that many problems in statistical mechanics and QFT center on the analysis of integrals of the form: \...
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0answers
66 views

Snoidal wave solutions of the $\phi^4$ model

I want to prove the existence of snoidal wave solutions of the $\phi^4$ model, given by $$u_{tt}-u_{xx}=u-u^3,\; (x,t) \in \mathbb{R}\times \mathbb{R}.$$ So, we are looking for solutions in the form $...
17
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1answer
556 views

Anomaly in QFT physics v.s. determinant line bundle

In a quantum field theory (QFT) lecture, a math-physics professor explains the anomaly in physics, say the non-invariance of the partition function of an anomalous theory under background field ...
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0answers
43 views

Composition of operators in $w_{1+\infty}$ and $W_{1+\infty}$

The algebra $W_{1+\infty}$ can be defined as a central extension of the lie algebra $w_{1+\infty}$ (defined as being spanned by $\left(-\partial_z \right)^m z^{-k}$ ). See for example: Alexandrov, ...
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0answers
104 views

How do I evaluate the following double integral?

I would like to evaluate the following double integral: $$ \int_{-1}^1d\zeta\int_{-1}^1 d\bar{\zeta} (\zeta+\bar{\zeta})^{d-2}[(1+\zeta\bar{\zeta})(\zeta-\bar{\zeta})]^J \,\times [(1-\zeta)(1+\bar{\...
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1answer
109 views

Renormalization group strategies

Before introducing block spin transformations in chapter four of Random Walks, Critical Phenomena and Triviality in Quantum Field Theory, the authors state the following: "In this chapter we sketch ...
4
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2answers
410 views

QFT and its notations

I know hardly anything about quantum field theory (QFT) but I'm giving a try to understand some ideas of it. As far as I understand, in QFT one is interested in studying measures such as: \begin{...
4
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0answers
94 views

Relationship between canonical commutation relations and projective representations?

$\DeclareMathOperator\CCR{CCR}\DeclareMathOperator\Im{Im}\DeclareMathOperator\PU{PU}$Let $V$ be a real vector space equipped with an antisymmetric bilinear form $\omega$. Recall that there is a $C^\...
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0answers
89 views

Spins in classical statistical mechanics

I'm reading Kupiainen's notes on the renormalization group and also caught my attention. Actually, this is something that often causes my some confusion. On page 43, in the section about Ginzburg-...
6
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1answer
229 views

Fourier transform on Minkowski space

Physicists Some people like to define the "Fourier transform" on Minkowski space as $\hat f(\xi) = \int e^{i \eta(x,\xi)} f(x) dx$, where $\eta(x,\xi)$ is the Minkowski form. I'm used to thinking of ...
2
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0answers
136 views

Decoding Fock spaces

Technically, the Fock space is (the Hilbert space completion of) the direct sum of the symmetric or antisymmetric tensors in the tensor powers of a single-particle Hilbert space H.(Wikipedia) ...
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80 views

What are the tempered Gibbs measures of classical $\phi^4$-theory?

I consider classical $\phi^4$-theory on the lattice. The model is defined in finite volume with Hamiltonian \begin{align*} H(\phi) = - \sum_{x \sim y} J_{x,y} \phi_x \phi_y \end{align*} and a-priori ...
36
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7answers
3k views

Interpretation of the action in classical mechanics

In classical mechanics the dynamics on a manifold $M$ are characterised by the minimisation of a functional $$ \min_{q \in C^\infty(\mathbb{R},M)} \int_{\mathbb{R}}L(q(t),\dot{q}(t))dt, $$ where $L:TM\...
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0answers
38 views

Long-time evolution under Schroedinger equation followed by short-time free evolution

Suppose $\psi_t(x)$ solves the Schroedinger equation $$ i \partial_t\psi_t=H\psi_t$$ where $$H= -\Delta +V.$$ Fix $s>0$. Suppose $t$ is large. Is it true that under certain conditions, the vectors ...
6
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2answers
213 views

Movement of repelled particles in a ball

EDIT: Given a system of $N\geq 3$ charged point particles in $\mathbb{R}^3$ of the same charge which interact according to Coulomb law (thus they repell one from each other). Is it possible that ...
1
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1answer
145 views

Square-integrable unbounded function

In R.D. Richtmyer, Principles of Advanced Mathematical Physics, p.85 an example is given of a continuous and square-integrable on $\bf{R}$ function, which is not bounded at infinity: $$f(x)=x^2\exp{(−...
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0answers
266 views

Lax pair of an integrable non-linear PDE

The following is a fourth-order non-linear PDE that passes the Painleve integrability test $$\left(1+x^{2}\right)^{2}u_{xxxx} + 8x\left(1+x^{2}\right)u_{xxx} + 4\left(1+3x^{2}\right)u_{xx}+ t\left(...
1
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1answer
161 views

Fourier transform of a translation invariant operator on $l^2(\mathbb{Z}) \otimes l^2(\mathbb{Z})$

Consider the space $l^2(\mathbb{Z}) \otimes l^2(\mathbb{Z})$ with distinguished computational basis $e_i \otimes e_j $ and a group of translations $T_a$ defined by $T_a e_i \otimes e_j = e_{i+a} \...
1
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1answer
145 views

The derivative of a filter with respect to a output singal [closed]

I have two signals, $d(t)$ and $p(t)$, respectively the input and the output of the matching filter $w(t)$, i.e. $$ d(t)*w(t)=p(t) $$ where $*$ denotes convolution.The impulse response $w(t)$ may be ...
1
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0answers
75 views

L2 norm of the diagonal entries of a random rotation of a fixed matrix?

Let $X\in\mathbb{R}^{d\times d}$ be the diagonal matrix with $d/2$ entries equal to $1$ and $d/2$ entries equal to $-1$. Let $F_U \triangleq \frac{1}{d}\|\operatorname{diag}(U^{\dagger}XU)\|^2_F$ ...
8
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4answers
2k views

mathematical physics without partial derivatives

Remark: All the answers so far have been very insightful and on point but after receiving public and private feedback from other mathematicians on the MathOverflow I decided to clarify a few notions ...
2
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0answers
32 views

Solving equations of motion of holomorphic BF theory - pure gauge in complex coordinates

In this paper by Bailieu and Tanzini, aspects of holomorphic BF theory are presented. Holomorphic BF theory on a four dimensional Kahler manifold is discussed from page 5, and on page 8 the ...
2
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0answers
57 views

Eight-dimensional hypersurface related to $SU(3)$

$\lambda_i$, $i=1,\ldots 8$ being Gell-Mann matrices, symmetric $d_{ijk}$ and antisymmetric $f_{ijk}$ $SU(3)$-tensors are defined through the multiplication law (see https://projecteuclid.org/euclid....

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