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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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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votes
0
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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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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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vote
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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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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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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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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 :
$$
\begin{array}{|c|c|c|c|c|}
\hline 0 & \varepsilon & 2 \...
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1
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Deriving Lorentz-covariant expression for the retarded Green's function of wave equation in $n+1$ dimensions
Consider spacetime to be homogeneous and isotropic. Then, the Green's function for the wave equation satisfies
\begin{equation}
\square G(x^{\mu}) = \delta^{(n+1)}(x^{\mu}).\tag{1}
\end{equation}
In $...
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Explaining the DHR superselection theory through specific examples
The DHR superselection theory is an important result in the framework of algebraic quantum field theory that categorizes the set of all physically admissable superselection sectors of an observable ...
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How to relate mathematicaly rigorous spinor fields to the ones used in physics?
One way to rigorously define spinor fields on metric manifolds is through the language of associated bundles. Namely, we have a principal bundle $P \overset{\pi}{\longrightarrow} M$ over $\mathrm{Spin}...
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Is there a character ring for quantum groups?
It is a well known fact that for any (reasonable) group $G$, the character ring and the representation ring are isomorphic,
$$
\chi_{R_1}(g)\chi_{R_2}(g)=\chi_{R_1\otimes R_2}(g),\qquad g\in G
$$
Is ...
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2
answers
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Is it possible to explain the Ultraviolet Catastrophe as a manifestation of the Riemann-Lebesgue Lemma?
Is it possible to explain the Ultraviolet Catastrophe as a manifestation of the Riemann-Lebesgue Lemma?
I don't fully understand any of both topics, but reading about the Ultraviolet Catastrophe on ...
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From the spectral theorem to the completeness relation in quantum mechanics
I often heard that the eigenfunctions of a Hermitian operator form a completeness basis, as
$$
\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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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
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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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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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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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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 ...
- 1,260
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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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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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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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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
$$U(q_a,q_b;T)= $$
$$\bigg(\...
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Intuition for gauge force as a 2-form
I understand the 2-form of a principle bundle as a measurement of the difference in horizontal lifts. In other words, if the lie bracket of two vector fields on the base does not equal to the lie ...
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votes
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Perturbation Method: What is the acceptable method to terminate expansion
I am using the book Classical Dynamics of Particles and Systems by STEPHEN T. THORNTON, JERRY B. MARION, page: 67 and they use perturbation method to approximate:
\begin{equation} T = \frac{kV + g}{...
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Explicit form of Dirac field creation/annihilation operators?
The explicit form of the creation and annihilation operators for the complex scalar field seems to be shown in all QFT lectures notes, but not those for the Dirac field (instead they tend to only give ...
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Is $\hat X^n$ a good observable?
Let $T:D(T)\subset L^2(\mathbb{R})\to L^2(\mathbb{R})$ be a linear operator defined as integer power of position operator
$$T:=X^n, \quad n\in\mathbb{Z}$$
Has it got any self-adjoint extensions? I'm ...
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In what way is conditional quantum probability restrictive, and why?
This is close to a duplicate of https://mathoverflow.net/q/412327/ but with a different emphasis. Unlike the mathoverflow equivalent, here I want to ask for your informed intuition as physicists.
To ...
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Associativity of Commutators inside bra-ket [duplicate]
I am having problem to see the contradiction of the following working:
Consider two arbitrary operators $\hat A$, $\hat B$, such that $[\hat A, \hat B ] =
c\hat1$, where $c$ is a non-zero constant. ...
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0
votes
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Classical Green function
What is the physical reason why the classical Green's function is not defined as a principle value integral?
In a recent discussion (Classical Green's function) it was said that the classical ...
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Fourier decomposition range in field quantization procedure
Consider the complex Klein-Gordon field (in finite volume $V$), which can be expanded in terms of plane waves as:
$$
\phi\left(\mathbf{x},t\right)=\frac{1}{\sqrt{V}}\sum_{\mathbf{k}}\left(A_{\mathbf{k}...
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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 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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Boundary condition of continuity for a barrier of 1D wave function with variable effective mass
When dealing with potentials, quantum wells, etc. I've usually used the following conditions for assuring the continuity of a wave function:
$$1. \ \psi_I(x)|_{x=0} = \psi_{II}(x)|_{x=0}$$
$$2. \ \...
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Are there any situations in physics when uniform convergence shouldn't be ignored?
A uniformly convergent series $k_n(x)$ can be multiplied or differentiated term-by-term:
$\sum_{n=0}^{\infty} \int k_n dx \equiv \int \sum_{n=0}^{\infty} k_n dx$ and
$\sum_{n=0}^{\infty} \frac{d}{dx} ...
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Moduli space for Riemann surfaces with boundaries and open string loop diagrams
I'm searching for information on the moduli space for Riemann surfaces with boundaries, like the ones used to compute open string loop diagrams. I found a huge lot of info for the case without ...
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In what sense does a pure spinor represent the orientation of a unique spacelike codimension-2 plane?
References 1 and 2 define a pure spinor $\psi$ to be a solution of the Cartan-Penrose equation
$$
\newcommand{\opsi}{{\overline\psi}}
v^\mu\gamma_\mu\psi=0
\hspace{1cm}
\text{with}
\hspace{1cm}
v^\mu\...
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Existence of informationally complete POVM on infinite dimensional Hilbert space
Do there exist informationally complete POVM (in this sense) on infinite dimensional separable Hilbert spaces?
It is known that these exist in finite dimensions (I believe they were used in the proof ...
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Complex, Holomorphic connection, and Symplectic — a not-so-Kähler manifold?
While studying mathematical physics, I wondered whether if there is a mathematical theory for a manifold that
has a complex structure [almost complex structure $J^\mu{}_\nu$ with vanishing Nijenhuis ...
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Path Integral for Fokker-Planck equation
As per Wio, the special case of the Fokker-Planck equation (in SDE form)
\begin{equation*}
dX = f(x)dt + \sqrt{2D} dW_t
\end{equation*}
has the path integral representation in the Ito scheme as
\...
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Wightman QFTs corresponding to Minimal Models
Per suggestion, I am cross-posting the following question from MathOverflow (original). At MO, Abdesselam referenced the work [2]. Glancing through it, is not immediately obvious to me how far the ...
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Why is the Chern number an integer?
I have relatively limited knowledge on the Chern number, and I know that there exists high-level math proofs that the Chern number is an integer, but let me try to focus on the case I have in mind. $\...
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Observables that encode all the information in a wavefunction
Let's consider the position space representation of the single particle Hilbert space and for simplicity let's stick to one dimension: $L^2(\mathbb{R})$. Let's say a collection of observables $O_1,...,...
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Mathematically rigorous QFT text
There are multiple questions on here about QFT textbook recommendations, but I am looking for mathematically precise texts on QFT.
Recommendations of introductory and advanced texts are welcome, but ...
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What is the name of this construction of quantum things?
Let $E$ be a set, and let $\mathcal{F}$ be a set of maps $E \rightarrow \mathbb{R}$ that "carries all the information on $E$", that is, the map $x \in E \mapsto (f(x))_{f \in \mathcal{F}}$ ...
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Can we determine when the lowest-energy state cannot be annihilated by any local operator, just by inspection of the Hamiltonian?
Relativistic quantum field theory (QFT) has the property that the lowest-energy state cannot be annihilated by any operator that is localized in a finite region of space (references 1,2,3). In other ...
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Is the momentum operator self-adjoint on any bounded interval on $\mathbb{R}$?
I know that the momentum operator $p$ defined on the Schwarz function $S(\mathbb{R})$ is essentially self-adjoint. However, what if I were to restrict $p$ to $C_c^\infty (0,1)\subseteq L^2(0,1)$. In ...
- 1,924
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Is this definition of the Fourier Transform of a quantum field operator rigorous?
Let there be a a quantum field operator $\hat\phi(t,\vec{x})$ which, because it acts (pointwise) on a separable Hilbert space, I expect I can write as
$$\hat\phi(t,\vec{x}) = \sum_n\sum_m\phi^n_m(t,\...
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