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How algebraic geometry and motives appears in physics?

First, I'm not a physicist so I have just a little background in physics. I have been reading some noncommutative geometry books and papers (Connes, Rosenberg, Kontsevich etc) and a lot of high ...
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851 views

Quivers in String Theory

Why do a physicist, particularly a string theorist care about Quivers ? Essentially what I'm interested to know is the origin of quivers in string theory and why studying quivers is a natural thing ...
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892 views

The derivation of fractional equations

Recently I saw some physical problems that can be modeled by equations with fractional derivatives, and I had some doubts: is it possible to write an action that results in an equation with fractional ...
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438 views

What are orbifolds and why are they useful and interesting for physics?

Just what the title says. What's the basic definition of an orbifold? How do they arise in physics and why are they interesting?
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544 views

Algebraic/Axiomatic QFT vs Topological QFT

Can anybody please tell me a good source investigating the relation between Algebraic/Axiomatic Quantum Field Theory (AQFT) and Topological Quantum Field Theory (TQFT)? Or is there none?
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Is causality a formalised concept in physics?

I have never seen a “causality operator” in physics. When people invoke the informal concept of causality aren’t they really talking about consistency (perhaps in a temporal context)? For example, if ...
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353 views

What is the algebraic property that corresponds to a topological term?

Warning: This question will be fairly ill-posed. I have spent a lot of time trying to make it better posed without success, so please bear with me. A single $SU(2)$ spin may be represented by the ...
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In coordinate-free relativity, how do we define a vector?

Relativity can be developed without coordinates: Laurent 1994 (SR), Winitzski 2007 (GR). I would normally define a vector by its transformation properties: it's something whose components change ...
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“An operator is hermitian”. Implications?

Alastair Rae states that there are 4 postulates of Quantum Mechanics in his text on the subject matter. The first part of his second postulate can be stated as: Every dynamical variable may be ...
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625 views

Why are the inner products of the eigenfunctions of an operator with a discrete eigenvalue spectrum guaranteed to exist?

I was reading through a textbook, and the statement was made that the inner products are guaranteed to exist if the eigenvalue spectrum of the operator is discrete. I have come across no support for ...
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649 views

The vacuum in quantum field theories: what is it?

In Section 10.1 of his textbook Quantum Field Theory for Mathematicians, Ticciati writes Assuming that the background field or classical source $j(x)$ is zero at space-time infinity, the presence ...
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346 views

Fourier Methods in General Relativity

I am looking for some references which discuss Fourier transform methods in GR. Specifically supposing you have a metric $g_{\mu \nu}(x)$ and its Fourier transform $\tilde{g}_{\mu \nu}(k)$, what does ...
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940 views

Why does iteratively solving the Hartree-Fock equations result in convergence?

[ Cross-posted to the Computational Science Stack Exchange: http://scicomp.stackexchange.com/questions/1297/why-does-iteratively-solving-the-hartree-fock-equations-result-in-convergence ] In the ...
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Why isn't the path integral defined for non homotopic paths?

Context In the Aharonov Bohm effect, there is a solenoid which creates a magnetic field. Since the electron cannot be inside the solenoid, the configuration space is not simply connected. Question ...
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EM wave function & photon wavefunction

According to this review Photon wave function. Iwo Bialynicki-Birula. Progress in Optics 36 V (1996), pp. 245-294. arXiv:quant-ph/0508202, a classical EM plane wavefunction is a wavefunction (in ...
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1answer
209 views

Time derivative of the state vector as expressed in abstract Hilbert space vs. as a wavefunction

The Schrodinger equation in Hilbert space is expressed as : $$\frac{\partial}{\partial t} \psi(t) = \frac{-i}{\hbar}H\psi(t). $$ Here $\frac{\partial}{\partial t} \psi(t) \equiv \psi'(t) \equiv\lim ...
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What are the implications for the AdS/Cft program if AdS is unstable?

To my understanding recent progress in the study of the non linear stability of AdS spacetime suggest that $AdS$ might be unstable. If this is true, what are the physical and mathematical ...
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692 views

Representations of Lorentz Group

I'd be grateful if someone could check that my exposition here is correct, and then venture an answer to the question at the end! $SO(3)$ has a fundamental representation (spin-1), and tensor product ...
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4answers
495 views

Why must the field equations be differential?

In Landau–Lifshitz's Course of Theoretical Physics, Vol. 2 (‘Classical Fields Theory’), Ch. IV, § 27, there is an explanation why the field equations should be linear differential equations. It goes ...
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Schwinger representation of operators for n-particle 2-mode symmetric states

A bosonic (i.e. permutation-symmetric) state of $n$ particles in $2$ modes can be written as a homogenous polynomial in the creation operators, that is $$\left(c_0 \hat{a}^{\dagger n} + c_1 ...
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607 views

Discrete gauge theories

I'm trying to understand a particular case of gauge theories, namely discrete spaces on which a group G can act transitively, with a gauge group H which is discrete as well. From what I've already ...
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1answer
393 views

Integral representation of Thomas-Fermi Equation

The Thomas-Fermi equation with dimensionless variables is identified as; $$ \frac{d^2\phi}{dx^2} = \frac{\phi^{3/2}}{x^{1/2}} $$ with the boundary conditions as $$ \phi(0) = 1 \\ \phi(\infty) = 0. $$ ...
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534 views

Transforming a sum into an integral

I posted this in the mathematical forums. Maybe you will help me. I found an hard article http://prola.aps.org/abstract/PR/v105/i3/p776_1 of yang huang and luttinger. The authors begins with the sum: ...
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165 views

Making new sense of the three-body problem in the light of Maryam Mirzakhani math contributions

I am unfamiliar with moduli spaces and ergodic theory which appear to be essential in Maryam Mirzakhani's math contributions which won her the Fields Medal. However, I am well conversant in essential ...
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369 views

Noether currents for the BRST tranformation of Yang-Mills fields

The Lagrangian of the Yang-Mills fields is given by $$ \mathcal{L}=-\frac{1}{4}(F^a_{\mu\nu})^2+\bar{\psi}(i\gamma^{\mu} D_{\mu}-m)\psi-\frac{1}{2\xi}(\partial\cdot A^a)^2+ ...
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604 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 ...
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467 views

Classical mechanics: Generating function of lagrangian submanifold

I have a short question regarding the geometrical interpretation of the Hamilton-Jacobi-equation. One has the geometric version of $H \circ dS = E$ as an lagrangian submanifold $L=im(dS)$, which is ...
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145 views

Does the complex 3-sphere have a complex structure modulus?

This question has a flavor which is more mathematical than physical, however it is about a mathematical physics article and I suspect my misunderstanding occurs because the precise mathematical ...
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2answers
451 views

Lie algebra and Lie group about quantum harmonic oscillator

We know that in the quantum harmonic oscillator $H=a^\dagger a$, $a^\dagger$, $a$, $1$ will span a Lie algebra, where $a, a^\dagger$ are the annihilation and creation operators, and $H$ is the ...
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3answers
636 views

Impressions of Topological field theories in mathematics

There have been recent results in mathematics regarding Conformal field theories and topological field theories. I am curious about the reaction to such results in the physics community. So I guess a ...
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3answers
335 views

Is particle number a problem for formulating statistical physics in a mathematically rigorous manner?

Quantities like the chemical potential can be expressed as something like $$\mu=-T\left(\tfrac{\partial S}{\partial N}\right)_{E,V}.$$ Now the entropy is the log some volume, which depends on the ...
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4answers
574 views

Using supersymmetry outside high energy/particle physics

Are there applications of supersymmetry in other branches of physics other than high energy/particle physics?
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283 views

Are there rigorous constructions of the path integral for lattice QFT on an infinite lattice?

Lattice QFT on a finite lattice* is a completely well defined mathematical object. This is because the path integral is an ordinary finite-dimensional integral. However, if the lattice is infinite, ...
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3answers
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Group Cohomology and Topological Field Theories

I have a two-part question: First and foremost: I have been going through the paper by Dijkgraaf and Witten "Group Cohomology and Topological Field Theories". Here they give a general definition for ...
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2answers
493 views

Is the step of analytic continuation unavoidable or can you model around it?

One sometimes considers the analytic continuation of certain quantities in physics and take them seriously. More so than the direct or actual values, actually. For example if you use the procedure ...
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3answers
140 views

“tmf$(n)$ is the space of supersymmetric conformal field theories of central charge $-n$”

I read this intriguing statement in John Baez' week 197 the other day, and I've been giving it some thought. The post in question is from 2003, so I was wondering if there has been any progress in ...
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339 views

Question about associative 3-cycles on G2 manifolds

Let $X$ be a manifold with $G_2$ holonomy and $\Phi$ be the fundamental associative 3-form on $X$. Let $*\Phi$ be the dual co-associative 4-form on $X$. Now consider a particular associative 3-cycle ...
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3answers
944 views

Use of 'complete' as in 'complete set of states' or 'complete basis'

Question. In the context of QM, I hear the phrases 'complete set of states' and 'complete basis' (among other similar expressions) thrown around rather a lot. What exactly is meant by 'complete'? ...
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1answer
858 views

Divergent Series

Why is it that divergent series make sense? Specifically, by basic calculus a sum such as $1 - 1 + 1 ...$ describes a divergent series (where divergent := non-convergent sequence of partial sums) ...
8
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1answer
258 views

Rank of the Poincare group

There are two Casimirs of the Poincare group: $$ C_1 = P^\mu P_\mu, \quad C_2 = W^\mu W_\mu $$ with the Pauli-Lubanski vector $W_\mu$. This implies the Poincare group has rank 2. Is there a way to ...
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1answer
588 views

AGT conjecture and WZW model

In 2009 Alday, Gaiotto and Tachikawa conjectured an expression for the Liouville theory conformal blocks and correlation functions on a Riemann surface of genus g and n punctures as the Nekrasov ...
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4answers
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Why can't any term which is added to the Lagrangian be written as a total derivative (or divergence)?

All right, I know there must be an elementary proof of this, but I am not sure why I never came across it before. Adding a total time derivative to the Lagrangian (or a 4D divergence of some 4 ...
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2answers
246 views

Deriving the expectation of $[\hat X,\hat H]$

For a free particle of mass $m$, with Hamiltonian $$\hat{H} = \frac {\hat{P}^2} {2m},$$ where $$\hat{P} = -i \hbar \frac{\partial} {\partial x}.$$ The commutative relation is given by $$[\hat{X}, ...
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294 views

When motion begins, do objects go through an infinite number of position derivatives?

This might be a very vague and unclear question, but let me explain. When an object at rest moves, or moves from point $A$ to point $B$, we know the object must have had some velocity (1st derivative ...
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1answer
161 views

Relation between cohomology and the BRST operator

Given a manifold $M$, we may define the $p$th de Rham cohomology group $H^p(M)$ as the quotient, $$C^p(M) \, / \, Z^p(M)$$ where $C^p$ and $Z^p$ are the groups of closed and exact $p$-forms ...
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305 views

Motivation for the Deformed Nekrasov Partition Function

I have recently been doing research on the AGT Correspondence between the Nekrasov Instanton Partition Function and Louiville Conformal Blocks (http://arxiv.org/abs/0906.3219). When looking at the ...
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615 views

Nonseparable Hilbert space

What kind of things can go wrong if we try to do quantum mechanics on a nonseparable Hilbert space? I have heard that usual mathematical manipulations that we take for granted will no longer hold. ...
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625 views

A confusion from Weinberg's QFT text (a vanishing term in Lippmann-Schwinger equation)

I was reviewing the first few chapters of Weinberg Vol I and found a hole in my understanding in page 112, where he tried to show in the asymptotic past $t=−∞$, the in states coincide with a free ...
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1answer
53 views

Sub and super multiplicativity of norms for understanding non-locality

In relation to various problems in understanding entanglement and non-locality, I have come across the following mathematical problem. It is most concise by far to state in its most mathematical form ...
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1answer
611 views

Iterated dimensional regularization

Given a 2-loop divergent integral $\int F(q,p)\,\mathrm{d}p\mathrm{d}q$, can it be solved iteratively? I mean I integrate over $p$ keeping $q$ constant Then I integrate over $q$ In both iterated ...