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6

The principle of the superposition of quantum states, or, as I shall refer to it, the sum over the alternatives, holds for particles belonging to a multiply-connected space in the same way as it holds for particles belonging to a simply-connected one, since it is one of the fundamental principles of quantum theory. On the other hand, what must be better ...


4

OP wants to evaluate $$ {\rm Tr}(-)^F e^{-\beta H}=\int_{PBC}[d\phi][d\psi] e^{-S_E[\phi,\psi]},\tag{2.5} $$ in Ref. 1 with periodic boundary conditions (PBC) for both the boson $\phi\equiv x$ and the fermion $\psi$. One can assume that the corresponding Fourier components are labelled by integers $n\in\mathbb{Z}$. One may argue that (2.5) does not depend ...


3

You can find an explanation of scalar fields and associated quantum effects in the Schwarzschild background in chapter four of these lecture notes. The article also contains references which might be of use to you.


3

There is no better definition than what Wikipedia offers - in general, a topological excitation is a (field) state, i.e. a localized quantity since fields depend on spacetime, whose integral is a topological invariant. One prime example are Yang-Mills theories in 4D, where the integral $\int \mathrm{Tr}(F\wedge F)$, as essentially the second Chern class of ...


3

As showed by Solovay here, in a non-separable Hilbert space $H$ there may be probability measures that cannot be written, for any $M$ closed subspace of $H$, as $\mu (M)=\mathrm{Tr}[\rho \mathbb{1}_M]$, for some positive self-adjoint trace class $\rho$ with trace 1 (density matrix). Here $\mathbb{1}_M$ denotes the orthogonal projection on $M$. [The proof of ...


2

$\vec{A}$ is the vector potential. $\vec{E}$ is equal to $-\nabla\phi$ only in the electrostatic case. If the electric field varies with time, we have \begin{equation} \nabla\times\vec{E} = -\frac{\partial\vec{B}}{\partial t} \end{equation} Since $\nabla\cdot\vec{B}$ is always zero, $\vec{B}$ is written as a curl of another field, called the vector ...


2

I usually see it in the reverse way, but it is a matter of taste. Hilbert spaces, in general, can have bases of arbitrarily high cardinality. The specific one used on QM is, by construction, isomorphic to the space L2, the space of square-integrable functions. From there you can show that this particular Hilbert space is separable, because it is a theorem ...


2

When you introduce an auxiliary variable, such as a regularization parameter, at the end of the calculation you have to take the limit that sets the expression back to the original one. If you introduce multiple auxiliary variables, you have to do this for all of them. Otherwise you're just doing a different integral. In this case specifically, ...


2

The overall idea is the following. You have a state evolving with the full interacting theory $\Psi(t) = U_t\Psi$. If the theory admits an asymptotic description for $t\to -\infty$, there must be a state $\Psi_0$ evolving with the free theory $\Psi_{0}(t) = U_0(t)\Psi_0$ such that the two evolutions ``coincide'' at large time in the past: $$\lim_{t\to ...


1

Well, the general theory is that you define the generalized wave operators for any couple of self-adjoint operators $A$ and $B$ to be: $$\Omega^{\pm}=\mathrm{s-lim}_{t\to \mp\infty} e^{iAt}e^{-iBt}P_{ac}(B)\; ,$$ where $\mathrm{s-lim}$ stands for the limit in the strong operator topology, and $P_{ac}(B)$ is the projection on the absolutely continuous ...


1

I believe my answer here to the question "How does one determine ladder operators systematically?" gives at least a partial answer to your question. It is a partial answer because I assume a little more than your bare question, but then, as we see by looking carefully at yuggib's answer observes you can obviously write down a hamiltonian with equally spaced ...


1

$\nabla\times\vec{A}$ is the solenoidal component of the vector field: it is the divergenceless component. A good way to intuitively visualise the Helmholtz theorem is to think in Fourier space, so that all fields become their Fourier transforms. In this visualisation, $\nabla\times \vec{E}$ is the component of the Fourier transform that is at right angles ...



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