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.
1,592
questions
2
votes
0answers
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
votes
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 ...
0
votes
0answers
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 $...
0
votes
0answers
86 views
Physics applications of quantum logic
Are there any examples of quantum logic being applied to solve actual physical questions, in particular to predict the physical properties (spectrum etc.) of some quantum-mechanical system? (Note that ...
1
vote
0answers
24 views
How to proceed in this Boundary value problem where Eigen values are calculated numerically?
While solving a boundary value problem (background provided in the Context section) I reach the following variable separated two equations ($F(x)$ and $G(y)$)
\begin{eqnarray}
\lambda_h F''' - 2 \...
-1
votes
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
votes
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
votes
0answers
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.
$\...
11
votes
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}...
1
vote
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 ...
3
votes
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 $S^3$ ...
5
votes
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
votes
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 ...
4
votes
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 ...
7
votes
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
votes
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
votes
0answers
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}\...
2
votes
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 ...
0
votes
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 \\
...
2
votes
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
votes
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 ...
1
vote
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
votes
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 ...
8
votes
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 ...
1
vote
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 ...
0
votes
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 $C_1$),
\...
4
votes
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
votes
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:
\...
1
vote
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
votes
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 ...
2
votes
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, ...
2
votes
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{\...
5
votes
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
votes
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
votes
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^\...
1
vote
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
votes
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
votes
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)
...
8
votes
0answers
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
votes
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, $$
where $L:TM\...
0
votes
0answers
38 views
Long-time evolution under Schroedinger equation followed by short-time free evolution
Suppose $\psi_t(x)$ solves the Schroedinger equation $$ i \partial_t\psi_t=H\psi_t$$ where $$H= -\Delta +V.$$
Fix $s>0$. Suppose $t$ is large. Is it true that under certain conditions, the vectors ...
6
votes
2answers
213 views
Movement of repelled particles in a ball
EDIT:
Given a system of $N\geq 3$ charged point particles in $\mathbb{R}^3$ of the same charge which interact according to Coulomb law (thus they repell one from each other). Is it possible that ...
1
vote
1answer
145 views
Square-integrable unbounded function
In R.D. Richtmyer, Principles of Advanced Mathematical Physics, p.85 an example is given of a continuous and square-integrable on $\bf{R}$ function, which is not bounded at infinity:
$$f(x)=x^2\exp{(−...
6
votes
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
$$\left(1+x^{2}\right)^{2}u_{xxxx} + 8x\left(1+x^{2}\right)u_{xxx} + 4\left(1+3x^{2}\right)u_{xx}+ t\left(...
1
vote
1answer
161 views
Fourier transform of a translation invariant operator on $l^2(\mathbb{Z}) \otimes l^2(\mathbb{Z})$
Consider the space $l^2(\mathbb{Z}) \otimes l^2(\mathbb{Z})$ with distinguished computational basis $e_i \otimes e_j $ and a group of translations $T_a$ defined by $T_a e_i \otimes e_j = e_{i+a} \...
1
vote
1answer
145 views
The derivative of a filter with respect to a output singal [closed]
I have two signals, $d(t)$ and $p(t)$, respectively the input and the output of the matching filter $w(t)$, i.e.
$$
d(t)*w(t)=p(t)
$$
where $*$ denotes convolution.The impulse response $w(t)$ may be ...
1
vote
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
votes
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
votes
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
votes
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....
