# Questions tagged [mathematical-physics]

DO NOT USE THIS TAG for elementary physical questions. This tag is intended for questions on modern mathematical methods used in quantum theory, general relativity, string theory, integrable system etc at an advanced undergraduate or graduate level.

3,626 questions
Filter by
Sorted by
Tagged with
34 views

### How to set up Chern-Simons theory, compute its "topological gauge shift" term, and compute its quantum path integrals with Wilson lines?

Suppose we want to consider Chern-Simons theory on an (odd-dimensional) compact boundaryless smooth manifold $X$ for a Lie group (the "structure/gauge group") $G$. Is it possible to do so ...
1 vote
16 views

23 views

### Defining free Dirac operator on $\big(L^2(\mathbb{R}^n)\big)^{n+1}$

So I have been considering free Dirac operator $D_0:(L^2(\mathbb{R}^3))^4 \supset (W^{2,1}(\mathbb{R}^3))^4 \to (L^2(\mathbb{R}^3))^4$ given by formula \begin{equation} D_0 = \left[\begin{matrix} ...
1 vote
66 views

### Are differential equations beyond coordinates?

In Physics when we write down Newton's second law, the differential equation we have is coordinate agnostic. Meaning, we can put in any coordinates into the equation and get a second order DE which ...
15 views

### How to derive the following approximation for a set of coupled equations?

Background I have the following set of coupled equations $\frac{d}{dt}\rho_{ab} = -i\omega A - \frac{1}{2}\Lambda B$ with $\omega, \Lambda$ parameters that are real and $A, B$ 4x4 matrices that are ...
21 views

### what are vassiliev invariants and how are they related to Chern-Simmons?

Possibly this question is very open for this platform and a little diffuse but I am a physicist and my knowledge in tides is not that high but I do not understand graphically what the vassiliev ...
1 vote
18 views

### 2n independent variables and DOF = n in Hamiltonian Mechanics

I’m studying Lagrangian and Hamiltonian mechanics for the first time. In the Lagrangian approach, that is the one I’ve studied first, a fundamental point is the definition of the degree of freedom of ...
23 views

### What does it mean for the momentum operator P to satisfy P|n>=hn|n>?

Using periodic boundary conditions to analyze momentum eigenstates on the circle $S^1$, we have the momentum operator $P=-i\hbar\frac{d}{d\phi}$ which satisfies $P|n>=\hbar n|n>$ (for clarity, |...
40 views

1 vote
43 views

### Integral measure quantum fields

In many physics books (especially QFT), the integral measure for a quantum field is written down as \begin{equation} \phi(x)=\int\frac{d^{3}p}{(2\pi)^{3}}.... \end{equation} It is a summation over the ...
38 views

### A boundary value problem of a Harmonic potential

A 2D electrostatic (i.e. harmonic potential) boundary value problem is shown in the figure. The solid lines are conductors (all are parallel), the two conductors with potential $V$ are infinitely long,...
34 views

### Error in Exponential Sum Formula

I have noticed that, for the exponential sum formula: $$\sum_{n=0}^{N-1} r^n = \frac{1-r^N}{1-r}.$$ the formula is not correct (left side $\neq$ right side) when N has a factional part (not just ...
29 views

### Time evolution and matrix ODE

Let $M$ be a linear operator on a finite-dimensional complex Hilbert space (thus, $M$ is just a matrix). Assume that $\text{Tr}M = 1$, $M$ is self-adjoint and $M \ge 0$ (positive semi-definite). In ...
20 views

### Why does the solution to pendulum problem with the geometric approach of Jacobi metric does not correspond to the solution with Lagrangian approach?

When we solve the pendulum problem with EL equation, we get to the differential equation $\ddot{q}+\frac{g}{l}\sin q=0$ but when I apply the substitution $t \rightarrow t\sqrt\frac{g}{l}$ and ...
1 vote
32 views

### Change orientation of turns of a trajectory

Consider a robot following the dynamical system $R_B'=R_B\exp(\omega_B^{\times}), v_W'=R_B\alpha_B + g_W$, where $R_B\in SO(3), v, \alpha, \omega, g \in \mathbb{R}^3$, and $\omega^\times$ is the skew ...
102 views

### Do the "physicist common knowledge" that "solenoidal vector fields have closed integral curves" have any mathematical foundation?

I remember having heard some physicist claiming that the integral curves of the magnetic field have to be closed, or "closed at $\infty$", due to the fact that the magnetic field is ...
41 views

### Find $\max_V \text{Tr} \left((Z_2 (V \otimes I) Z_1 (V^\dagger \otimes I)\right)$

I am doing a quantum optimization where the final problem has the following form $$\max_V \text{Tr} \left((Z_2 (V \otimes I) Z_1 (V^\dagger \otimes I)\right),$$ where $V \in \mathbb{C}^{d\times d}$ is ...
46 views

1 vote
49 views

### What kinds of questions are people asking and answering with 𝐸8 lie algebra? [closed]

I don't understand much because there are many papers about it but I don't know what the objectives are or what is sought with this group of lie? or where is the investigation going or what is it ...
83 views

### Calculate Projectile flight time based on Gravity

Hey I want to calculate the flight time of a Projectile in 3D Space based on Bullet's speed, Velocity, Acceleration and Gravity or a custom downward force. I already have a formula that calculates the ...
24 views

### How to determine the fractal dimension of a random walk in 2D?

I have to determine the fractal dimension of a random walk in 2D. How can I do that? I am first supposed to picture the random walk for 2D as in the following site. However, I am also asked to ...