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0answers
46 views

(Causal) Set notation round brackets vs square brackets? [on hold]

In many (quite old) papers & books I have been reading recently in the causal theory of general relativity (e.g. On the structure of causal spaces, Kronheimer & Penrose, 1967) I find sets ...
1
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0answers
34 views

What is the charge density for line and surface charges?

In electrostatics it is common to see line, surface and volumetric charges being described differently. A line distribution is a function defined on the line, a surface charge distribution is a ...
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0answers
32 views

physics thermofluid mechanics pressure force area [on hold]

As shown in the attached image, a tank has an inclined wall at an angle of $45^{\circ}$ to the horizontal. On this wall, there is a 1 m square door that is hinged at A and has a simple latch at B. The ...
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1answer
45 views

Thermofluid mechanics inclined plane

As shown in the attacked image, a tank has an inclined wall at an angle of 450 to the horizontal. On this wall, there is a 1m square door that is hinged at A and has a simple latch at B. The distance ...
0
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0answers
18 views

Construct recurrence relation for the temporal evolution of a Master equation

Say that we have a system evolving over discrete timesteps. The quantity we are interested is X and is given by a distribution $P_X$. This distribution is evolving temporally, and we have a ...
1
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1answer
38 views

Homoclinic orbit and a particle in a double well

The physical set-up is a classical particle in a parabolic double well: Physically, a particle with reasonable amount of potential energy would be able to roll down the slope of the well, roll past ...
1
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0answers
29 views

Is there a Bayesian theory of deterministic signal? Prequel and motivation for my previous question

This is a prequel to my question: What's the probability distribution of a deterministic signal or how to marginalize a dynamical system? Clearly my question looks at the same time fairly ...
1
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0answers
69 views

What's the probability distribution of a deterministic signal or how to marginalize dynamical systems?

In many signal processing calculations, the (prior) probability distribution of the theoretical signal (not the signal + noise) is required. In random signal theory, this distribution is typically a ...
0
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0answers
25 views

Simplify $K$-matrix

2+1D Abelian topologically ordered states are believed to be described by multicomponent $U(1)$ Chern-Simons theories, with Lagrangian \begin{equation} ...
1
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1answer
68 views

When and where to check the formal definition of a manifold

In most texts on GR we are first introduced to a formal and rigorous definition of a manifold. We then learn the point that in GR "any coordinate system" might be used for the 4D spacetime metric. ...
0
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2answers
46 views

What's meant by: “All observers agree on the combination of basis vectors and components for a tensor”?

There's a Youtube video given by Dr Dan Fleisch on Tensors, where he states at 11:22: You may be wondering, what is it about the combination of components and basis vectors that makes tensors so ...
3
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1answer
38 views

Generators of a certain symmetry in Quantum Mechanics

In Classical Mechanics to describe symmetries like translations and rotations we use diffeomorphisms on the configuration manifold. In Quantum Mechanics we use unitary operators in state space. We ...
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0answers
42 views

Extending projection operator to infinite-dimensional case

Hi I have a basic question regarding bra-ket notation. Given that $\{|e_n \rangle \}$ is a discrete orthonormal basis, $$\langle e_m | e_n \rangle = \delta_{mn}$$ then $$\sum_{n}|e_n \rangle \langle ...
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0answers
102 views

How much analysis do we need to define a (classical) quantum system?

I'm trying to figure out to what extent is analysis involved in the definition of a (classical) quantum system. Classical meaning not QFT. Here's a definition I came up with: Defnition: A quantum ...
0
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1answer
44 views

Allowed Wave Functions of System

Given a single-particle system with Hamiltonian $H$, what constraints can be put on the wave function at a particular point in time $\psi(x)$? Of course $\psi(x)$ must obey boundary conditions given ...
1
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0answers
36 views

Lippmann-Schwinger equation and time dependence

Consider the Lippmann-Schwinger equation (LSE) $$ |\psi\rangle = |\phi\rangle + \hat{G}_0(\epsilon) \hat{V} |\psi\rangle \tag{1}$$ where $\hat{G}_0(\epsilon) = \frac{1}{\epsilon - \hat{H}_0 + ...
4
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0answers
55 views

Interpreting the Fourier transform of a Gibbs measure

Recall that a Gibbs measure gives a probability distribution on states $x$ of the form $$ p_X(x) = \frac{1}{Z(\beta)}\exp(-\beta E(x)) $$ As I understand, the function $E$ is interpreted as the ...
2
votes
0answers
32 views

Contour integral of the retarded Klein Gordon propagator

I've been trying to prove by hand the Peskin's formula for the retarded propagator of the Klein Gordon equation, that is, $$\int_{x^0 > y^0} \frac{d^4p}{(2\pi)^4} \frac{-e^{-ip(x-y)}}{i(p^2 - ...
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votes
1answer
53 views

Cross-differentiation to derive the maxwell relation $\left(\frac{\partial T}{\partial V}\right)_S = -\left(\frac{\partial P}{\partial S}\right)_V$ [closed]

How can I use $T=\left(\frac{\partial E}{\partial S}\right)_V$ and $P=-\left(\frac{\partial E}{\partial V}\right)_S$ to derive $$\left(\frac{\partial T}{\partial V}\right)_S = -\left(\frac{\partial ...
0
votes
3answers
66 views

A relationship between entropy and temperature

I tried deriving a formula relating entropy (not change in entropy, but entropy itself) to temperature. I’ve only seen two equations really relating to entropy thus far, and only one of them includes ...
1
vote
1answer
58 views

QM sytem with eigenvalues of the form $f(m*n)$ and prime number gap spectrum

Depending on the dimension and the symmetry and form of the potential, the energy eigenvalues of a quantum mechanical system have different functional forms. Eg. The particle in the 1D-box gives rise ...
1
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0answers
37 views

Are physical functions always differentiable [duplicate]

I know that physicist usually don't really think too much about differentiabillity of functions. Usually there are at most finite many points where functions aren't differentiable and if there are ...
3
votes
1answer
61 views

Equivariant cohomology and Mayer-Vietoris sequence

I'm reading this article upon topological field theory and I'm a bit confused about the way he compute equivariant cohomology of $S^2$ wrt $\mathrm{U}(1)$, i.e. $H^\bullet_{S^1}(S^2)$. You can find ...
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0answers
17 views

Potential of an axisymmetric disc with constant rotation velocity

I am having trouble understanding why the form of the 3D potential for a disc with a constant rotation velocity for circular orbits of stars within the disc \begin{equation} v(R) = v_0, ...
1
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1answer
42 views

Book to study Dirac delta function from a physics point of view [duplicate]

I am a beginning physics graduate student. I am often bewildered by the strange properties of the Dirac delta function such as: $\delta (a x)= \frac{1}{a} \delta (x)$ The derivative of $\delta (x)$ ...
1
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0answers
36 views

Sufficient condition for square integrability [duplicate]

The necessary condition for $\int\limits_{-\infty}^{+\infty}|\psi(x)|^2dx$ to be integrable is that $\psi(x)\rightarrow 0$ as $x\rightarrow\pm\infty$. But this is not the sufficient condition. For ...
6
votes
2answers
181 views

Implementing Category Theory in General Relativity

I was thinking if it may be possible to implement category theory in general relativity. I don't mean writing simply in terms of categories, but actual fundamental ideas (i.e. physics of the theory ...
1
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0answers
39 views

Understanding the mathematical terms in Physics [closed]

I understand well only when I get a physical picture of the concept. Is it good? or should I change.
4
votes
2answers
152 views

When does causal separation imply no spacelike separation?

(See here for notation.) In Minkowski space, if $p\prec q$, then there is no spacelike curve $c:[0,1]\to \mathbb{R}^{n-1,1}$ with $c(0)=p$ and $c(1)=q$. This is obvious from a spacetime diagram. Here ...
3
votes
1answer
36 views

Pointwise temperature and Thermodynamic Temperature

When I studied Thermodynamics the definition of temperature I've learned (based on the Gibbs' postulates) was: $$T = \dfrac{\partial U}{\partial S}.$$ This quantity has all the properties of the ...
4
votes
0answers
202 views

How does one determine if a spacetime is globally hyperbolic?

A spacetime $M$ is said to be globally hyperbolic if it is strongly causal and if the sets $J^+(p)\cap J^-(q)$, for all $p,q\in M$, are compact. (For more information, see the Wiki article on causal ...
2
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0answers
43 views

Seeking masters degree advice [closed]

I am graduating this semester with a bachelors in physics. My goal is to do theory in my graduate level course work. The problem I am in now is that I missed the deadline to take the physics GRE (I'm ...
0
votes
1answer
42 views

Assumptions in physics for Helmholtz decomposition

A version of the Helmholtz theorem says that, under opportune assumptions on the vector field $\boldsymbol{F}:\mathbb{R}^3\to\mathbb{R}^3$ and on $V\subset\mathbb{R}^3$ the following identity holds: ...
0
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0answers
44 views

About the formula of pendulum simple

for the modulation and the simulation of a pendulum simple , I'm Find this formula : a(t) = a0 * sin ( sqrt(g/l) * t * Pi/2 ) - [ k/(mll) * cos ( sqrt(g/l) * t * Pi/2 ) * t ) ] ...
1
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1answer
83 views

Spectral functions in quantum mechanics

I'm a math student and a totally newcomer to quantum mechanics and I'm trying to teach myself this subject by studying Faddeev and Yakubovski's Lectures on Quantum Mechanics for Mathematics Students. ...
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0answers
49 views

What are some resources for Algebraic quantum mechanics for Physicists

I am interested in the GNS construction and other stuff. I am aware of Valter Moretti's book but I want something that is more inclined towards physicists.
1
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0answers
49 views

Resources for Mathematical Physics [closed]

I intend to study mathematical physics (on my own or continuing education). Could you suggest resources (a bibliography of top papers, online courses, continuing education courses, channels/playlists, ...
3
votes
1answer
86 views

Is it possible to integrate a function over a null hypersurface?

For $f$ a compactly supported function on a spacetime $(\mathcal{M},g)$, one may define its integral over $\mathcal{M}$ by $$f\longmapsto \int_\mathcal{M}\star f,$$ where $\star$ is the Hodge star of ...
3
votes
3answers
88 views

Examples of Bernoulli Numbers, Euler-Mascheroni Integration, and the $\zeta(n)$ in physics [closed]

In Arfken's Mathematical Methods for Physicists, there is a subsection of the "Infinite Series" chapter which covers the Bernoulli numbers, Euler-Mascheroni integration (or summation), and the ...
1
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1answer
42 views

What is the relation between Hilbert space constructed from the GNS construction and the standard Hilbert space-state?

I recently started reading Algebraic quantum mechanics. So I have no knowledge of the subject. In the GNS construction we construct the Hilbert space of states as follows, We endow the algebra of ...
1
vote
2answers
88 views

Physicists definition of vectors based on transformation laws

First of all I want to make clear that although I've already asked a related question here, my point in this new question is a little different. On the former question I've considered vector fields on ...
2
votes
1answer
54 views

Is there some physical intuition behind Clifford Algebras?

The mathematically rigorous definition of a Clifford Algebra is as follows: Let $V$ be a vector space over a field $\mathbb{K}$ and let $Q : V\to \mathbb{K}$ be a quadratic form on $V$. A Clifford ...
1
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0answers
12 views

Ultrasound Transducers and Simulator [closed]

Presently I am working on Underwater Acoustic Wireless Transmission. I desire to measure water parameters at the bottom of the surface of the water and then pass it to the water surface using ...
2
votes
1answer
43 views

vorticity not zero - why

I am quite new to fluid dynamics and I cannot seem to understand the concept of vorticity. It is defined as $\nabla\times\vec{v}$ where $\vec{v}$ is the velocity vector. Now (working in cartesian ...
6
votes
1answer
104 views

Are path integrals integrals with countable or uncountable infinite dimensions?

Path integrals are integrals with infinite dimensions. But I recently became confused about if the number of dimensions are discrete/countable or continuous/uncountable. I always thought it should be ...
2
votes
0answers
38 views

What are the essential pure mathematics branches applied in theoretical high energy physics [closed]

I am a physics graduate student.I am interested in theoretical high energy physics.Very often people say that to be a good theoretical physicist you need to know mathematics very well.Now although we ...
1
vote
2answers
83 views

Two particle system state space

I'm trying to understand how the state space of a bigger system composed of smaller subsystems relates to the state spaces of the individual subsystems. To get started I'm currently trying to ...
1
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1answer
65 views

Difference between local inertial frame and coordinate chart

In the most cases the local inertial frame is definied "physically" but I'm searching for a mathematically meaningful definition of the local inertial frame to solve my problem: Is the local ...
1
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0answers
45 views

Is there an introduction to non-commutative geometry for non-physics and mathematics students?

I am looking for a simple explanation as how spectral triples give rise to definition of distance using Dirac operators?
3
votes
2answers
98 views

What is the significance of diagonal matrices in Quantum Mechanics?

I'm currently taking quantum mechanics and diagonal matrices, along with the idea of diagonalization, comes up alot. One line in my textbook threw me off completely and made me realize I don't ...