Questions tagged [mathematical-physics]

DO NOT USE THIS TAG just because your question involves math! If your question is on simplification of a mathematical expression, please ask it at math.stackexchange.com. Mathematical physics is the mathematically rigorous study of the foundations of physics, and the application of advanced mathematical methods to problems in physics. Examples include partial differential equations (PDEs), variational calculus, functional analysis, and potential theory.

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Random matrix theory and the singularities of the Weingarten function

In the random matrix theory literature, one often encounters identities associated with averages over ensembles of random unitaries. For a simple example let's say we're interested exclusively in $2\...
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Numerical solution of 2D Wave Equation using Fourier Transform and Finite Difference

This is the 2D Wave equation: $$ u_{tt} = u_{xx} + u_{yy} $$ using initial condition: $u(x,y,0)=f(x,y), \:\: u_{t}(x,y,0) = 0$. The inverse Fourier transform used is: $$ u(x,y,t) = \int\int \hat{u}(\...
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Path integral on many-body quantum mechanics

Suppose $\mathscr{H}$ is a Hilbert space describing a one-particle quantum system and $\mathcal{F}(\mathscr{H})$ is its associated Fock space, which is used to describe a many-body quantum system. Let ...
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5 votes
1 answer
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Question about the kinetic energy operator

The Kinetic Energy Operator is essentially self-adjoint. Under what circumstances does it have a unique extension?
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9 votes
1 answer
286 views

Resolution of the identity of operator with mixed spectrum

In most quantum mechanics text books, the resolution of the identity or completeness relation is stated in the following (or similar) form $$ \mathbb I_\mathcal H = \sum\limits_n |\lambda_n\rangle \...
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Representations of minimal model primary fields in the Coulomb-gas Formalism

This question is a cross-post from MO (link). Is it known how to construct the primary field operators of the unitary minimal models $\mathcal{M}(m+1,m)$ in the Coulomb gas formalism? As far as I can ...
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Why does the variational approach become a lower bound as the number of replicas approach zero

Consider solving a disordered system with Hamiltonian $H[h(x)]$ where $h(x)$ denotes the disorder parameter/random variable at lattice site $x$ (e.g., possibly of independent Gaussian distributions $\...
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Is higher or lower specific heat better for cooling an object via contact?

Suppose you want to cool an object by putting it into contact with another object, much colder object, and the transferal of joules to an intermediary equilibrium temperature is instantaneous. If this ...
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Inequality constraint in Lagrangian

An example for an equality constraint: \begin{equation} x \geq x_a \end{equation} which can be used in the lagrangian: \begin{equation} \mathcal{L} = E(x) + \lambda(x-x_a) \end{equation} but ...
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Uncertainty of waves

Here in the pictures I have written about some question I have been thinking about a long time, what do you think? Link to the chapter I am talking about: http://www.its.caltech.edu/~matilde/...
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Why is Dirac's Phase Operator Non-Hermitian?

I'm self-studying Gerry and Knight. To prove Dirac's phase operator is non-existent, the book makes the following argument. The conventions used are as follows: $\hat{n}$ is the number operator, $\hat{...
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Electron density via bosonization/refermionization

I'm currently trying to understand the rigorous construction of bosonization/refermionization via Jan von Delft. In the constructive approach, we consider a system on a finite $L$ circle and thus in ...
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What does it mean to divide by $dx$? (e.g. in thermo. equations) [duplicate]

There are many relations between partial derivatives that are useful in thermodynamics. Some are listed in appendix A.5 of Callen's Thermodynamics and an introduction to thermostatistics. When trying ...
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General derivation of Supervirasoro algebra

I'm looking for a derivation derivation of the ($\mathcal{N} = 1$) Supervirasoro algebra (NS sector) that does't just apply to specific examples. Most books/papers either just cite the result, or ...
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2 votes
1 answer
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Extracting $\mathbf b$ from $M = a I + \mathbf b \cdot \mathbf S$, when $S_i$ are higher spin matrices?

This is a cross-post of a question that I posted on the Math SE, that did not get any answers there. It is fundamentally a mathematics question, but it pertains to spin matrices, which many Physics SE ...
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1 answer
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Coupled Oscillator Period [closed]

I was studying an example of a coupled oscillator the other day, namely two identical masses attached to three springs, the lateral ones of which with the same elastic constant, when I came across the ...
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3 votes
1 answer
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Recommendations for Algebraic quantum mechanics book

I am familiar with quantum mechanics and quantum information at the level of Sakurai and Preskill's lecture notes / Nielsen and Chuang. I want to study the $C^*$ algebraic formulation of quantum ...
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Is a reasonable assumption to consider that the contact point of the Euler's Disk (with stationary center of mass) trace this finite bounded spiral?

Is a reasonable assumption to consider that the contact point of the Euler's Disk (with stationary center of mass) trace this finite bounded spiral? This question is highly related to working with the ...
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3 votes
1 answer
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Anti-holomorphic contribution to 2d conformal algebra

I am reading Ginsparg's notes on 2D-CFT, and I am deeply confused about why Ginsparg states after (1.8) that the conformal algebra for 2d Euclidean space consists of two copies of the Witt algebra. My ...
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1 vote
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How the Zero-point Energy of the System containing 2 Fermions in 3 Micro-Energy States is 1?

If we distribute 2 Fermions $\mathrm{(A,A)}$ in 3 Micro-Energy States (0,$\epsilon$,$2\epsilon$), the confirmation is given by : $$ \begin{array}{|c|c|c|c|c|} \hline 0 & \varepsilon & 2 \...
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4 votes
1 answer
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Deriving Lorentz-covariant expression for the retarded Green's function of wave equation in $n+1$ dimensions

Consider spacetime to be homogeneous and isotropic. Then, the Green's function for the wave equation satisfies \begin{equation} \square G(x^{\mu}) = \delta^{(n+1)}(x^{\mu}).\tag{1} \end{equation} In $...
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Explaining the DHR superselection theory through specific examples

The DHR superselection theory is an important result in the framework of algebraic quantum field theory that categorizes the set of all physically admissable superselection sectors of an observable ...
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How to relate mathematicaly rigorous spinor fields to the ones used in physics?

One way to rigorously define spinor fields on metric manifolds is through the language of associated bundles. Namely, we have a principal bundle $P \overset{\pi}{\longrightarrow} M$ over $\mathrm{Spin}...
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Is there a character ring for quantum groups?

It is a well known fact that for any (reasonable) group $G$, the character ring and the representation ring are isomorphic, $$ \chi_{R_1}(g)\chi_{R_2}(g)=\chi_{R_1\otimes R_2}(g),\qquad g\in G $$ Is ...
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Is it possible to explain the Ultraviolet Catastrophe as a manifestation of the Riemann-Lebesgue Lemma?

Is it possible to explain the Ultraviolet Catastrophe as a manifestation of the Riemann-Lebesgue Lemma? I don't fully understand any of both topics, but reading about the Ultraviolet Catastrophe on ...
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7 votes
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From the spectral theorem to the completeness relation in quantum mechanics

I often heard that the eigenfunctions of a Hermitian operator form a completeness basis, as $$ \sum_i | i \rangle \langle i | = \hat{1} \tag{1} $$ and the mathematical foundation is the spectral ...
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(In)finite lattice in quantum statistical mechanics: validity of phase classifications and TQFT [closed]

I would like to understand the motivation for studying quantum statistical mechanics, such as spin models, on an infinite lattice, or in other word, in the operator algebraic framework. I learned that ...
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5 votes
1 answer
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Examples of quantum systems modelled with Type II von Neumann algebras

What are the examples of quantum systems that should be modelled with a Type $II_1$ or $II_\infty$ von Neumann algebra? I am pretty much a novice at von Neumann algebra, so I have hard time finding ...
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-2 votes
2 answers
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Do analytical results always "trump" numerical ones? [closed]

Suppose I have a system that can be described by some differential equation(s). If I can manage to write down a proper analytical solution to it, but which I can't quite replicate numerically, whether ...
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3 votes
3 answers
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Is there a mathematically rigorous formalization of "operator-valued vectors" from quantum mechanics? [duplicate]

I've seen in various quantum mechanics courses people define various "operator-valued vectors" for the case of three-dimensional systems. For example, people define momentum as $\hat{\vec{p}}...
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6 votes
1 answer
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Example of GNS construction

I can't find any pedagogical and illustrative example of "step-by-step" GNS construction in the literature. What I mean by illustrative? - writing explicitly the functional $\rho: A \...
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12 votes
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Is it known whether Wightman's axiomatic QFT is logically equivalent to Osterwalder–Schrader's axiomatic QFT?

Constructive QFT has provided some interesting models for dimension $d < 4$ of space-time, satisfying specific axiomatic versions of QFT. On the other hand, it is a well known fact that an ...
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2 votes
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Quantum Fields living in finite dimensional non-unitary irreducible representations of the Lorentz group

In Non-unitary, finite dimensional representations of the Lorentz group it got clarified that the finite dimensional non-unitary reps of the Lorentz group are completely reducible. In physics, we use ...
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Single particle space for the free Euclidean field

In Quantum Field Theory, the free field of mass $m$ can be constructed from creation and annihilation operators on the Fock space. Let $\mathscr{H}_1$ be the single-particle Hilbert space, $F(\mathscr{...
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4 votes
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Extension to excited states of Lieb's Theorem for the Hubbard model

Lieb's theorem shows that for the Hubbard model, $$\hat{H} = -t \sum_{ \langle \mu,\nu \rangle, \sigma} \hat{c}^\dagger_{\mu \sigma}\hat{c}_{\nu \sigma} + U \sum_\mu \hat{n}_{\mu \uparrow}\hat{n}_{\mu ...
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3 votes
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Momentum operator in Geometric Quantization vs momentum operator on arbitrary curved space(time)s

In the following stack exchange post Momentum Operator in curved spacetime (QFT) a general expression for the momentum operator is given for a Riemannian manifold $(M,g)$. Similarly, Frederic Schuller'...
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8 votes
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Path integral in QM vs QFT

On page 282 of Peskin and Schroeder discussing functional quantization of scalar fields, the authors use expression 9.12, the path integral in ordinary quantum mechanics $$U(q_a,q_b;T)= $$ $$\bigg(\...
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1 vote
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Intuition for gauge force as a 2-form

I understand the 2-form of a principle bundle as a measurement of the difference in horizontal lifts. In other words, if the lie bracket of two vector fields on the base does not equal to the lie ...
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3 votes
2 answers
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Perturbation Method: What is the acceptable method to terminate expansion

I am using the book Classical Dynamics of Particles and Systems by STEPHEN T. THORNTON, JERRY B. MARION, page: 67 and they use perturbation method to approximate: \begin{equation} T = \frac{kV + g}{...
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Explicit form of Dirac field creation/annihilation operators?

The explicit form of the creation and annihilation operators for the complex scalar field seems to be shown in all QFT lectures notes, but not those for the Dirac field (instead they tend to only give ...
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Is $\hat X^n$ a good observable?

Let $T:D(T)\subset L^2(\mathbb{R})\to L^2(\mathbb{R})$ be a linear operator defined as integer power of position operator $$T:=X^n, \quad n\in\mathbb{Z}$$ Has it got any self-adjoint extensions? I'm ...
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In what way is conditional quantum probability restrictive, and why?

This is close to a duplicate of https://mathoverflow.net/q/412327/ but with a different emphasis. Unlike the mathoverflow equivalent, here I want to ask for your informed intuition as physicists. To ...
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1 vote
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Associativity of Commutators inside bra-ket [duplicate]

I am having problem to see the contradiction of the following working: Consider two arbitrary operators $\hat A$, $\hat B$, such that $[\hat A, \hat B ] = c\hat1$, where $c$ is a non-zero constant. ...
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Classical Green function

What is the physical reason why the classical Green's function is not defined as a principle value integral? In a recent discussion (Classical Green's function) it was said that the classical ...
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1 vote
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Fourier decomposition range in field quantization procedure

Consider the complex Klein-Gordon field (in finite volume $V$), which can be expanded in terms of plane waves as: $$ \phi\left(\mathbf{x},t\right)=\frac{1}{\sqrt{V}}\sum_{\mathbf{k}}\left(A_{\mathbf{k}...
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0 votes
2 answers
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Angular momentum and precession in classical Hamiltonian (symplectic) mechanics

In Hamiltonian mechanics, angular momentum is a certain momentum map and a component of the angular momentum is the generator function of the action of a one-parameter subgroup of the rotation group $...
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What is the intuitive pictorial explanation of a conservative force's criteria of having zero curl and a value equal to the gradient of a potential?

A conservative force should be satisfying these two criteria. I want to understand the intuitive or pictorial form of why the criteria of only having zero curl not necessarily mean the force is ...
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11 votes
1 answer
841 views

What does mathematical consistency in QFT mean?

My question is more naive than Is QFT mathematically self-consistent? Just when people talk about the mathematical consistency of QFT, what does consistency mean? Do people want to fit QFT into ZFC? ...
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Does choice of renormalisation scheme affect the consequences of Haag's theorem?

So Haag's theorem means that the interaction and Hamiltonian picture are not equivalent. The reason seems to be that renormalization mixes interactions and free particles (ie self energy of a free ...
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Is the set of every renormalization group countable and finite?

Is the set of every renormalization group countable and finite? Suppose A is a renormalization group, and the elements of it compose of the set B. Is B the set countable and finite?
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