Hot answers tagged quantization
17
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 ...
12
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. ...
12
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 ...
12
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 ...
11
A classical$^1$ 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 ...
9
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 ...
8
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 ...
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
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 ...
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 ...
7
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 ...
6
Discrete eigenvalues for measurable observables in QM occur more broadly than just for bound states. For example angular momentum, whether orbital or spin, is always restricted to discrete eigenvalues for all systems, free and unbound as well as bound systems. It is the case, however, that discrete spectra is 'associated' with compact geometries. Thus the ...
6
Your example shows that you may use symmetry to get a Hamiltonian (which should be invariant) and for classification of its solutions: it is convenient to choose wavefunctions in a way that they form the basis of irreducible representations of the symmetry group.
To get the numbers you need to solve the equations, their symmetry is not enough. Symmetry may ...
6
This answer addresses the geometrical origin of the Bohr-Sommerfeld condition.
In geometric quantization, the additional structure required beyond the
symplectic data of the phase space is a polarization. The quantization spaces are constructed as spaces of polarized sections with respect a polarization.
The most "obvious" type of a polarization is the ...
6
i) First of all, the Dirac quantization rule
$$\tag{1} \frac{qg}{2\pi\hbar} ~\in~ \mathbb{Z} $$
for magnetic monopoles can be generalized to the Dirac-Zwanziger-Schwinger quantization condition
$$\tag{2} \frac{q_1g_2-q_2g_1}{2\pi\hbar} ~\in~ \mathbb{Z} $$
for dyons. (In a slight misuse of terminology, we shall in the following also include purely ...
6
There are several forms of discreteness in quantum theory. The simplest one is the discreteness of eigenvalues and the associated countable eigenstates. Those arise similarly to the discrete standing waves on a guitar string. The boundary conditions only allow certain standing waves that nicely fit into the enforced region in space. Even though the string is ...
5
The classical equations of motion are not affected by changing the Lagrangian
$$L \qquad \longrightarrow \qquad L' = L+ \frac{dF}{dt}$$
by a total time derivative. Put $F= -q_1 q_2$. Then
$$L' = -\frac{1}{2}(q_1^2 + q_2^2).$$
This Lagrangian $L'$ does not contain time derivatives, and thus there are no dynamics.
The classical equations of ...
5
The fundamental reason is that light waves at temperature T have a definite length scale, a typical wavelength, and classical electromagnetic theory has a scaling invariance which forbids such a scale from emerging. In the classical theory, the energy in electromagnetic waves just leaks into ever smaller distances.
The leaking can be understood by thermal ...
5
A modern treatment of this subject can be found in Segreev's book on the Kahler geometry of loop spaces also available online. This line of research started with the seminal work of Bowick and Rajeev: The holomorphic geometry of the closed bosonic string theory and $Diff S^1/S^1$ (Spires) (and independently Kirillov and Yuriev (Please see the reference in ...
5
First, quite generally, it is not true that the action $S$ is ever required to be a multiple of Planck's constant $h$. What quantum mechanics implies is pretty much exactly the opposite thing. The formulation of quantum mechanics (any quantum mechanical theory) that uses the action is the Feynman path integral where the action enters via the exponent
$$ ...
5
I think of 'quantize' as a verb that refers to converting the classical to the quantum picture. The Hamiltonian you wrote down is, after all, the classical one, which is not 'quantized'. Once you solve it by (1) writing down annihilation/creation operators, (2) finding that only particular wavefunctions are allowed, (3) numerically evaluated the energy ...
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