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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On which bundle do QFT fields live?
In QFT, there is a vector field of electromagnetism, usually notated by $A$, which transforms as a 1-form under coordinate changes. Since quantum fields are operator-valued, I thought it is a section ...
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Effects of Localized Medium Changes on Field Propagation
I've studied various theories related to fields. These theories often include equations describing how the activity of a source is transmitted to other locations. The properties of the medium ...
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Why we use trace-class operators and bounded operators in quantum mechanics?
The set of trace-class operators $\mathcal{B_1(H)}$ on the Hilbert space $\mathcal{H}$ is like the Banach space $l^1$, while the set of bounded operators $\mathcal{B_\infty(H)}$ is like the Banach ...
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Jensen's inequality on (super)operator exponential
Let us define the expectation value $\langle A\rangle_{\rho}$ of a superoperator $A$ over a density matrix $\rho$ as $(\rho, A(\rho))$, where the scalar product between operators reads $(O_1,O_2):= Tr[...
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Equality of Hilbert subspaces
If $A,B\in \mathscr{L_H}$ in the lattice of subspaces of a Hilbert space $\mathscr{H}$,
then is it always true that $$A\sqsubseteq B\ \&\ B\sqsubseteq A \implies A=B\ ~ ?$$
Or is there maybe an ...
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Operator systems in functional analysis & quantum mechanics: intuition
I saw this concept of operator systems in here but I am not sure if I want to get deep into it before knowing roughly what it is used for in, say, quantum information or quantum mechanics.
My very ...
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Are $\mathcal{PT}$-symmetric Hamiltonians dual to Hermitian Hamiltonians?
I was reading this review paper by Bender, in particular section VI where they show that, despite $\mathcal{PT}$-symmetric Hamiltonians not being hermitian, they can have a real spectra. They go on ...
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Angular momentum Lie algebra for infinite-dimensional Hilbert spaces
Let $V := \operatorname{span}{(J_1, J_2, J_3)}$ be a Lie algebra over the complex numbers such that
$J_1$, $J_2$, and $J_3$ are essentially self-adjoint operators on some Hilbert space $\mathcal{H}$.
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Electric field due to plane at constant potential and a cylinder with no flux on surface
There is a plate at a constant potential V and potential equal to zero far away. the problem is two-dimensional. For this case, the electric field lines will simply be straight lines. Now let there be ...
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Rigorous definition of the average value $\langle a^{*}(f)a(g)\rangle$ on Fock spaces for arbitrary states [duplicate]
One of the axioms of quantum mechanics states that a quantum state $\rho$ is a positive (hence self-adjoint) trace class operator with trace one. Given an observable $A$, the expected value of $A$ in ...
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Why does the finite trace of density matrix imply a discrete Schmidt decomposition? [closed]
In the paper defining an average Schmidt number for a particular entangled system, Law and Eberly say:
Because density matrices always have finite trace, the Schmidt decomposition is always discrete, ...
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What are the distinct mathematical formalisms of quantum mechanics?
Consider the physical theory called non-relativistic quantum mechanics. What are the distinct mathematical formalisms for this physical theory? That is, different mathematical frameworks for this ...
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Non-orientability in electromagnetism
I'm currently studying E&M and I have a question related to the mathematical formalism of the theory. Electrodynamics depends heavily on the divergence and Stokes's theorem which in their ...
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Concrete statement about QFT not being mathematically rigorous [duplicate]
It is often mentioned that QFT is ill-defined mathematically. I have seen this as stated that QFT can be defined on a lattice, but that it breaks down if the lattice spacing goes to zero. ...
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Elasticity theory: homogeneous deformations of a perfect lattice
I want to understand how the macroscopic (linear) elasticity theory emerges from the microscopic properties of matter. My question is about the model of the "perfect lattice", which is used ...
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