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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 ...
1
vote
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. ...
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 ) ] ...
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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, ...
1
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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.
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
87 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
vote
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 ...
9
votes
1answer
127 views

Resources for algebraic topology in condensed matter physics

I wanted to know if anyone had any good introductions on algebraic topology for the theoretical physicist? I am particularly interested in applications to condensed matter physics, but would be happy ...
11
votes
1answer
796 views

When can we take the Brillouin zone to be a sphere?

When reading some literatures on topological insulators, I've seen authors taking Brillouin zone(BZ) to be a sphere sometimes, especially when it comes to strong topological insulators. Also I've seen ...
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 ...
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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 ...
1
vote
1answer
247 views

Divergent path integral

What does it mean to have a divergent path integral in a QFT? More specifically, if $$\int e^{i S[\phi]/\hbar} D\phi (t)=\infty $$ What does this mean for the QFT of the field $\phi $? The field ...
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 ...
15
votes
5answers
2k views

What does a frame of reference mean in terms of manifolds?

Because of my mathematical background, I've been finding it hard to relate the physics-talk I've been reading, with mathematical objects. In (say special) relativity, we have a Lorentzian manifold, ...
1
vote
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
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 ...
11
votes
2answers
452 views

Are derivations of physical laws less important than the laws themselves? [closed]

The proportionality between the kinetic energy of gas molecules and temperature is a well-known result. This is usually shown by considering a cubical box containing an ideal gas, and postulating that ...
1
vote
1answer
42 views

generating function method for expectation values of operator products

Many times in quantum physics we face a problem of calculating expectation values of normal product operators. Assume we have a two level system with creation and annihilation operators $\hat{a}$ and ...
4
votes
0answers
62 views

Axiomatic QFT: Time-slice Axiom vs Transformation Properties

I am studying Wightman axioms and Haag–Kastler axioms for QFT from Haag's book "Local Quantum Physics". In both axiomatic frameworks, he introduces the "Time-slice Axiom" (axiom G) as "There should ...
9
votes
2answers
357 views

Motivation to introduce von Neumann algebras in addition to $C^*$algebras?

Observables are self-adjoint elements of a $C^*$algebra. As such, this structure seems sufficient to describe physics. A theorem by Gelfand and Naimark says that a $C^*$algebra can always be ...
0
votes
2answers
39 views

Question about limit cycles and linear systems

In here http://users.isy.liu.se/en/rt/claal20/SysBio2015/Notes_SysBio_2015_partC.pdf it says: A limit cycle is however an intrinsically nonlinear concept: a linear system cannot have a limit ...
5
votes
2answers
434 views

Time-ordered operator in Srednicki

On page 51 Srednicki states, "Note that the operators are in time order...we can insert $T$ without changing anything". This I agree with. But then on the next paragraph he states "The time order ...
49
votes
5answers
5k 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 ...
15
votes
1answer
2k views

Onsager's Regression Hypothesis, Explained and Demonstrated

Onsager's 1931 regression hypothesis asserts that “…the average regression of fluctuations will obey the same laws as the corresponding macroscopic irreversible process". (Here is the links to ...
5
votes
0answers
98 views

Uniqueness of solution in newtonian mechanics

Recently I came across the problem of Norton's dome. I thought of two questions, for which I found no answer. Does there exist a newtonian initial value problem, where the total force on each body ...
5
votes
1answer
82 views

Can I *always* decompose a normalizable function into the discrete Hydrogen spectrum?

This question has been bothering me for a while now: can one reconstruct an arbitrary (normalizable) function $\phi(\mathbf r)$ in $\mathbb R^3$, with only the discrete set of Hydrogen wavefunctions ...
3
votes
1answer
122 views

Number theoretic loophole allows alternative definition of entropy?

A bit about the post I apologize for the title. I know it sounds crazy but I could not think of an alternative one which was relevant. I know this is "wild idea" but please read the entire post. ...
1
vote
1answer
136 views

Why is the logarithm of the number of all possible states of a system differentiable?

Temperature of a system is defined as $$\left( \frac{\partial \ln(\Omega)}{ \partial E} \right)_{N, X_i} = \frac{1}{kT}$$ Where $\Omega$ is the number of all accessible states (ways) for the system. $ ...
10
votes
1answer
789 views

String theory from a mathematical point of view

I have a great interest in the area of string theory, but since I am more focused on mathematics, I was wondering if there is any book out there that covers mathematical aspects of string theory. I ...
42
votes
6answers
5k views

Why does calculus of variations work?

How does it make sense to vary the position and the velocity independently? Edit: Velocity is the derivative of position, so how can you treat them as independent variables? Doesn't every physics ...
2
votes
2answers
68 views

Variant of the Sokhotski–Plemelj theorem

I am aware of the Sokhotski–Plemelj theorem (I have also heard people referring to it as the "Dirac identity") which states that in the limit $\eta\rightarrow 0^+$ $$\frac{1}{x\pm i\eta}=\mathcal ...
9
votes
3answers
952 views

Boundary layer theory in fluids learning resources

I'm trying to understand boundary layer theory in fluids. All I've found are dimensional arguments, order of magnitude arguments, etc... What I'm looking for is more mathematically sound arguments. ...
69
votes
7answers
8k views

Number theory in Physics [closed]

As a Graduate Mathematics student, my interest lies in Number theory. I am curious to know if Number theory has any connections or applications to physics. I have never even heard of any applications ...
2
votes
1answer
86 views

Conservation of momentum in infinite square well

This is inspired by Griffiths QM section 2.2, on the infinite square well, which is about how far I've gotten (so, sorry if this is addressed later in the book). For any given starting wavefunction, ...
3
votes
1answer
78 views

Semi-infinite forms?

I am reading Vafa's paper 'Topological Mirros and Quantum Strings'. In this paper, the author says the Hilbert Space of a fermionic string theory corresponds to the space of semi-infinite forms on the ...
3
votes
1answer
48 views

Do all equillibrium points of a discrete mapping show up on the bifurcation diagram?

The question in the title is perhaps vaguely posed, so I'll include the concrete example which is bugging me. Suppose we have a mapping given by $$N_{t+1}=N_t\cdot ...
4
votes
2answers
113 views

Radial quantum number for infinite circular well

For completeness, I will sketch the solution of a particle in an infinite circular well first and then get to my question. I apologize in advance since the introduction is standard undergraduate ...
5
votes
2answers
222 views

What is the correct relativistic distribution function?

General Statement and Questions I am trying to figure out the proper way to model a velocity/momentum distribution function that is correct in the relativistic limit. I would like to determine/know ...
0
votes
0answers
50 views

Tension in string

In the derivation of the 1d linear wave equation for small amplitude waves on 1d string, they said at the end that if the density $\rho(x)$ is assumed to be constant, then the Tension $T(t)$ would ...
1
vote
0answers
55 views

Physics : Formula involving density, Mass, temperature and pressure [closed]

A set of dryers has a mass 40kg and density of 16kg/m3 at temperature of 40°C under the pressure 760mmHg. At what temperature will the mass be 50kg with density 20kg/m3 under a pressure of 700mmHg
2
votes
1answer
79 views

What does unphyisical quantity mean? [closed]

I was reading this Terry Tao article and almost near the end he says "the terms involving infinity do not make particularly rigorous sense, but would be considered orthogonal to the application at ...
1
vote
1answer
55 views

response function and Fourier transform

A response function defined as the kernel of the following integral: $\rho(t) = \int_{-\infty}^t \chi(t,t') E(t')dt'$ (1), where $\chi(t,t')$ is the response function. Physically, it relates ...
2
votes
1answer
111 views

Derivation of Schwarzschild metric using the full machinery of differential geometry [closed]

How would one derive the Schwarzschild metric using the full machinery of differential geometry, using the component approach as little as possible? Something along these lines: Begin with a manifold ...