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4

By definition, a value is in the continuous spectrum of $A$ if it not an eigenvalue, but the range of $A-\lambda I$ is a proper dense subset of the Hilbert space. There is nothing in this definition distinguishing separable spaces, or precluding operators in them from having continuous spectrum, and indeed some do. There is an equivalent definition in terms ...


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Because these are actually Fourier transform of the usual Green functions. Considere the Schrödinger equation : $$ \hat{\mathcal{H}}|\Psi(t)\rangle=\mathrm{i}\partial_t|\Psi(t)\rangle $$ The general solution $|\Psi(t)\rangle$ of such equation for a time-independant hamiltonian $\hat{\mathcal{H}}$ can be expressed in terms of Green function $G(x',x,t)$ : $$ ...


2

There are a couple of points to be precise: in four spacetime dimensions there is no scalar relativistic interacting field theory that can be rigorously defined (i.e. in which the unitary dynamics can be constructed), at least for the moment. This does not mean it is not possible, but we have not the mathematical tools to do it. The physical calculations ...


2

The Fourier transform of a waveform has to contain an infinite range of frequencies in order to represent the full information content of the original. It is of course possible to create a waveform that only contains a finite range of frequencies (or rather - one for which the high frequencies vanish exponentially). The Gaussian is one such shape - its FT is ...


2

In principle it is not necessary to restrict the classical space to the set of solutions of the classical dynamics. The usual "basic" requirement for the classical space $X$ is the following: $(X,b)$ is a couple with $X$ a real vector space and $b:X\times X\to \mathbb{R}$ a non-degenerate skew-symmetric bilinear form (symplectic form). This is ...


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My recommendations are: For Mechanics, Mathematical Methods of Classical Mechanics, by Arnold. For Electromagnetism, Modern Electrodynamics, Zangwill. For Quantum Mechanics, Quantum Physics, Le Bellac, or, at an easier level, Introduction to quantum mechanics, Griffiths. For General Relativity, General Relativity, Wald. This should give you a (good) ...


2

Comments to the question (v4): First a disclaimer. Note that even for a smooth local functional, the existence of a functional/variational derivatives is not guaranteed, but depends on appropriately chosen boundary conditions. Calculus of variations, functional/variational derivatives, Frechet derivatives, Gateaux derivatives, etc, is a huge mathematical ...


2

Question: Does there exist a nontrival non-Legendre transformation T such that the function defined by F(q,p,t)=T[L(q,q˙,t)] contains the full dynamics of the system? Answer: any function that produces the equations of motion under some sort of rules that you state is an allowed function to describe the dynamics. In particular any function that you can ...


1

The question is really one of definition. In the math literature on self adjoint opertors the "discrete spectrum" is by definition that part of the spectrum which consists of normalizable states, while the "continuous spectrum" is that part where they are non-normalizable. It is possible to have a physical system (a random potential on the entrire real ...


1

I only had a brief look at the paper, but to me it looks like that even after choosing $A^{0}$, in such a way that $\nabla_{l}l=0$ and $l^{\mu}=(0,1,0,0)$ then you still have some freedom in what they call $\theta^0$ which will affect $m^{\mu}$ and $\overline{m}^{\mu}$ but not the first two. I will agree that there is no more freedom in $A^{0}$, only for ...


1

First of all, a small historical note. As far as I know, in constructive mathematics you can construct uncountable sets, e.g. the reals, since the usual way of introducing them (by Dedekind cuts) is constructible, i.e. involves a countable process. Of course it is not done in a finite number of steps method, but indeed constructible. Most of the people that ...


1

I'd like to point out that Springer has published a book on this subject. It really seems an interesting read: Leo Corry: David Hilbert and the Axiomatization of Physics (1898–1918), From Grundlagen der Geometrie to Grundlagen der Physik. $\quad \quad \quad \quad \quad \quad \quad \quad\quad $


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A quantum version of the Bochner's theorem, discussed for example by Bröcker and Werner and Srinivas and Wolf, gives a necessary and sufficient condition for the Wigner function (and P- or Q- functions which can be obtained from Wigner functions by convolution/deconvolution) to correspond to a valid density operator (or a positive operator in general) by ...


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I am not at all an expert of quantum gravity, but I think you have misunderstood the point. As I understand it, the point is not necessarily having distributions as quantum Hilbert space vectors, but having a distributional "configuration space", i.e. distributions as the domain of the function(al)s that are the quantum vectors. While in QM (i.e. for ...



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