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28

One more answer against “second quntization”, because I think it is a good demonstration of how a lame notation can obscure a physical meaning. The first statement is: there is no second quantization. For example, here is citation from Steven Weinberg's book “The Quantum Theory of Fields” Vol.I: It would be a good thing if ...


22

Gravity has to be subject to quantum mechanics because everything else is quantum, too. The question seems to prohibit this answer but that can't change the fact that it is the only correct answer. This proposition is no vague speculation but a logically indisputable proof of the quantumness. Consider a simple thought experiment. Install a detector of a ...


19

What you observe is the general phenomenon that in relativistic theories time translation is replaced by "affine-parameter-translation" or "wordline translation symmetry" and hence the corresponding Hamiltonian becomes a constraint, the constraint that states must be invariant under this symmetry. Yes, this works for the relativistic spinning particle and ...


17

Dear Calvin, if any portion of the world is described by probabilistic wave functions, then the whole world has to be. It's easy to show it. Take a decaying nucleus, connect it to a hammer that kills a cat a that also makes the Sun explode into 2 pieces. The nucleus is evolving into a linear superposition of "decayed" and "not yet decayed" states. ...


16

In a nutshell, the problem with OP's choice of operators $\hat{p}_j$ and $\hat{H}$ is that they are not selfadjoint wrt. to the pertinent measure $\mu$. In other words, the usual integration by part method to prove selfadjointness does not work. Here are some more details. Let us put the constants $m=1=R$ for simplicity. Then the Lagrangian reads $$\tag{1}...


15

Space and time are continuous, in quantum mechanics or otherwise. In particular, whenever our theories of any kind talk about time, it is always a real continuous parameter. Similarly, spatial positions of particles in ordinary quantum mechanics are operators $\hat x$ whose eigenvalues are continuous, too. This fact is related to the continuity of time by ...


15

A classical1 theory of (relativistic) $p$-dimensional membranes exists for any non-negative integer $p$. Such classical membrane-like objects appears in the full non-perturbative formulation of string theory. The problem arises if one tries to quantize a membrane (in a first quantized sense) using standard perturbative quantization methods, which have ...


14

First, you are right in that non-Minkowski solutions to string theory, in which the gravitational field is macroscopic, it should be thought of as a condensate of a huge number of gravitons (which are one of the spacetime particles associated to a degree of freedom of the string). (Aside: a point particle, corresponding to quantum field theory, has no ...


13

The ordering ambiguity is the statement – or the "problem" – that for a classical function $f(x,p)$, or a function of analogous phase space variables, there may exist multiple operators $\hat f(\hat x,\hat p)$ that represent it. In particular, the quantum Hamiltonian isn't uniquely determined by the classical limit. This ambiguity appears even if we require ...


13

Same as @SamRoelants answer but not restricted to gravitational waves. Given $$G_{\mu\nu}=8\pi T_{\mu\nu}$$ $T_{\mu\nu}$ is constructed from the matter fields (Klein Gordon, Dirac or whatever). These are operators (or operator-valued distributions if you like), hence so is the gravitational source $T_{\mu\nu}$. So the right hand side obeys the rules of ...


11

If I'm only allowed to use one single word to give an oversimplified intuitive reason for the discreteness in quantum mechanics, I would choose the word 'compactness'. Examples: The finite number of states in a compact region of phase space. See e.g. this Phys.SE post. The discrete spectrum for Lie algebra generators of a compact Lie group, e.g. angular ...


11

The general problem of converting classical expressions to quantum operator ones is in general unsolvable because classical mechanics is an approximation to quantum mechanics and not the other way around. There is always an ambiguity in how to order noncommuting operators. You have to handle it on a case by case basis, and there are a number of "quantization"...


10

The paper you quote covers a similar case, which was solved previously by S.T. Ma (Phys. Rev. 69 no. 11-12 (1946), p. 668), but deals with the scattering problem on the tail of the exponential - hence the complex energies. What follows is partly inspired by that paper but is quite distinct from it. The tricky part is not getting scared by the Bessel ...


9

The overall idea is the following. As the symplectic manifold is affine (in the sense of affine spaces not in the sense of the existence of an affine connection), when you fix a point $O$, the manifold becomes a real vector space equipped with a non-degenerate symplectic form. A quantization procedure is nothing but the assignment of a (Hilbert-) Kahler ...


8

You seem to be talking about the "old covariant quantization" in which $L_n$ for positive $n$ and $(L_0-a)$ annihilate physical ket states $|\psi\rangle$, right? It's analogous to the Gupta-Bleuler quantization http://en.wikipedia.org/wiki/Gupta-Bleuler_quantization which was a standard procedure used already in electromagnetism. The idea is that the ...


8

I have written an answer to Mathoverflow in which explicit formulas for the classical and quantum Hamiltonians of a spin system (Generators of $SU(2))$ were written explicitely. The classical Hamiltonians are given by means of functions on the two sphere and the quantum Hamiltonians by means of holomorphic differential operators (which act on the sections of ...


8

Concerning point c), on how complex numbers come into quantum theory: This has a beautiful conceptual explanation, I think, by applying Lie theory to classical mechanics. The following is taken from what I have written on the nLab at quantization -- Motivation from classical mechanics and Lie theory. See there for more pointers and details: Quantization ...


8

The word quantization surface is not standard terminology. It apparently refer to a (generalized) Cauchy surface. A (generalized) Cauchy surface is a hypersurface on which the initial conditions are given for a well-posed initial value problem. Phrased differently, for given initial conditions on the Cauchy surface, there exists a unique solution for the ...


8

The core of perturbative string theory has a mathematically rigorous formulation. In fact much of mathematical physics and mathematical insight into quantum field theory as such has been gained from the study of the low-dimensional QFTs that constitute the worldvolume theories of the string and the various branes. For instance the axiomatization of QFT in ...


8

It is possible to abstract the notion of a QFT away from the notion of Lagrangians/Hamiltonians, one axiomatic way are the Wightman axioms. As one can see, they reduce the quantum theory to its very heart: A Hilbert space where the states live and a field operator that acts upon it, generating "particles", all of this happening in a Lorentz covariant way. ...


8

Yes, one traditional alternative to the path integral formalism is the operator formalism. For QED with abelian gauge group, the old quantization formulation is the Gupta-Bleuler formulation. For QCD/Yang-Mills theory with non-abelian gauge group, the Gupta-Bleuler formulation is replaced by the BRST formulation. The BRST formulation exists in at least 3 ...


8

There doesn't exist any procedure to uniquely associate a Hermitian operator $L$ to a function of the phase space $f(x,p)$. Quantum mechanics is a theory that exists independently of classical physics. Quantum mechanics is not just a cherry on a classical pie that needs the classical theory to exist at every moment. If we want to define a quantum theory, we ...


7

One way to see the validity of the background field method (BFM) lies in the proof of the equivalence of the effective action calculated with the BFM to the standard effective action. Let $\Gamma[v]$ be the effective action (Legendre transform of the connected generating function $W[J]$) where $v=v(J)=\frac{\delta W[J]}{\delta J}$ is the "classical" field ...



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