nLab classical limit

Idea

To a large extent quantum physics is a deformation of classical physics parameterized by Planck's constant $\hbar$. For a given quantum system, the limit $\hbar \to 0$ (in as far as that can be made sense of), is called its classical limit.

One imagines that from a quantum field theory one can derive a classical field theory of which the original quantum field theory may be regarded as a quantization. Whereas quantization may be haphazard, since quantum physics seems to be a more fundamental description of reality that need not be derivable from anything classical, we expect taking the classical limit to be more systematic, since the appearance of a classical world in some regimes should be explicable by fundamental quantum theories. However, it may not be as systematic as some people would like, as in this blog discussion.

What precisely this means and to which degree precisely it makes sense depends on which precise formulation of these concepts one considers. For instance when regarding deformation quantization as definition the quantum theory then taking the classical limit is an obvious straightforward operation. But in more complete contexts, such as when the quantum field theory is realized as an FQFT or AQFT, the passage to the classical limit may be subtle or not make sense at all.

Most prescriptions for taking classical limits are formulated in algebraic settings. To fit for instance a 1-dimensional FQFT into the discussion one is likely to find in the literature, one will have to encode it equivalently in its spectral triple (as described there) and then proceed with that.

Formalizations

In various formalizations of (aspects of) quantum physics, the classical limit has a corresponding formalization. The following list discusses a few.

Relations to coherent states

One method for producing classical mechanics from a quantum theory is by looking at coherent states of the quantum theory. The standard (Glauber) coherent states have a localized probability distribution in classical phase space whose center follows the classical equations of motion when the Hamiltonian is quadratic in positions and momenta.

(For nonquadratic Hamiltonians, this only holds approximately over short times. For example, for the 2-body problem with a $1/r^2$ interaction, Glauber coherent states are not preserved by the dynamics. In this particular case, there are, however, alternative $SO(2,4)$-based coherent states that are preserved by the dynamics, smeared over Kepler-like orbits. The reason is that the Kepler 2-body problem – and its quantum version, the hydrogen atom – are superintegrable systems with the large dynamical symmetry group $SO(2,4)$.)

In general, roughly, coherent states form a nice orbit of unit vectors of a Hilbert space $H$ under a dynamical symmetry group $G$ with a triangular decomposition, such that the linear combinations of coherent states are dense in $H$, and the inner product $\phi^*\psi$ of coherent states $\phi$ and $\psi$ can be calculated explicitly in terms of the highest weight representation theory of $G$. The diagonal of the $N$-th tensor power of $H$ has coherent states $\psi_N$ (labelled by the same classical phase space as the original coherent states, and corresponding to the $N$-fold highest weight) with inner product

$$\phi_N^* \psi_N = (\phi^* \psi)^N$$

and for $N\to \infty$, one gets a good classical limit. For the Heisenberg group, $\phi^*\psi$ is a $1/\hbar$-th power, and the $N$-th power corresponds to replacing $\hbar$ by $\hbar/N$. Thus one gets the standard classical limit.

Formulation in geometric quantization

In the context of geometric quantization the classical limit corresponds to taking high tensor powers of the prequantum line bundle with itself. See at Planck's constant the section Planck’s constant - In geometric quantization.

References

Basic literature on relations between coherent states and the classical limit, based on irreducible unitary representations of Lie groups includes the book

• A. M. Perelomov, Generalized Coherent States and Their Applications, Springer-Verlag, Berlin, 1986.

and the paper

• L. Yaffe, Large $N$ limits as classical mechanics, Rev. Mod. Phys. 54, 407–435 (1982)

Both references assume that the Lie group is finite-dimensional and semisimple. This excludes the Heisenberg group, in terms of which the standard (Glauber) coherent states are usually defined. However, the Heisenberg group has a triangular decomposition, and this suffices to apply Perelomov’s theory in spirit.

• Roland Omnès, §6 of: The Interpretation of Quantum Mechanics, Princeton University Press (1994) [ISBN:9780691036694]

The online book

• Arnold Neumaier, Dennis Westra, Classical and Quantum Mechanics via Lie algebras (arXiv)

contains a general discussion of the relations between classical mechanics and quantum mechanics, and discusses in Chapter 16 the concept of a triangular decomposition of Lie algebras and a summary of the associated representation theory (though in its present version not the general relation to coherent states).

For other relevant approaches to a rigorous classical limit, see the online sources

• <http://www.projecteuclid.org/Dienst/Repository/1.0/Disseminate/euclid.cmp/1103859040/body/pdf>

• <http://www.univie.ac.at/nuhag-php/bibtex/open_files/si80_SIMON!!!.pdf>

• <http://arxiv.org/abs/quant-ph/9504016>

• <http://arxiv.org/pdf/math-ph/9807027>