Abstract
It is well known that the process of quantization—constructing a quantum theory out of a classical theory—is not in general a uniquely determined procedure. There are many inequivalent methods that lead to different choices for what to use as our quantum theory. In this paper, I show that by requiring a condition of continuity between classical and quantum physics, we constrain and inform the quantum theories that we end up with.
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Notes
As we will see later, I specifically propose the requirement that the weak topology on a quantum algebra of observables preserves the information encoded in the classical topology of pointwise convergence, which can be understood itself as a weak topology.
In particular, this rules out nonregular representations of the Weyl algebra.
One could restrict attention here to sequences because the norm topology is second countable, but for the weak topologies considered later, which are not second countable, one must work with arbitrary nets.
A complete normed vector space is called a Banach space. A C*-algebra is thus a Banach algebra whose norm is, in a certain sense, compatible with multiplication and involution.
A linear functional \(\rho \in {\mathfrak {A}}^*\) is positive if \(\rho (A^*A)\ge 0\) for all \(A\in {\mathfrak {A}}\) and normalized if \(||\rho || = 1\).
See Feintzeig (2016) for more on the completion of a C*-algebra into a W*-algebra.
We include complex-valued functions for generality. One typically restricts to the self-adjoint functions, which are real-valued, for describing observable quantities.
Of course there are other possible choices for the algebra of classical observables that one may use if one wanted to admit, e.g., unbounded observables.
Throughout this paper we work in units such that \(\hbar = 1\).
This definition captures as a special case the usual commutation relations
$$\begin{aligned}{}[{\mathcal {Q}}(q),{\mathcal {Q}}(p)] = i \end{aligned}$$for the quantized position observable \({\mathcal {Q}}(q)\) and the quantized momentum observable \({\mathcal {Q}}(p)\).
The Weyl algebra can also be uniquely characterized through its Hilbert space representations (see Clifton and Halvorson 2001).
The Weyl operators admit a unique maximal C*-algebra norm (Manuceau et al. 1974).
These are not the only interpretive positions available (see Ruetsche 2011). Albeit extreme views, they are illustrative for the purposes of this paper.
One gets this notion of pointwise convergence immediately for abelian C*-algebras of all continuous functions on a compact Hausdorff space (see Kadison and Ringrose 1997, p. 270). However, when we consider functions on a locally compact (not necessarily compact) manifold \({\mathcal {M}}\), the correspondence of weak convergence and pointwise convergence on \({\mathcal {M}}\) only holds for the algebra \(C_0({\mathcal {M}})\), rather than the algebra of all continuous functions, \(C({\mathcal {M}})\). The reason is that it is only for \(C_0({\mathcal {M}})\) that the pure states correspond to precisely the points in \({\mathcal {M}}\). Other algebras of bounded continuous functions allow for ‘states at infinity’—pure states that cannot be represented as points of \({\mathcal {M}}\)—which forces the weak topology to diverge from the topology of pointwise convergence on \({\mathcal {M}}\). Another way to state the main issue of this paper is as the question of whether these ‘states at infinity’ are physically significant.
On the other hand, if we started with a C*-algebra other than \(C_0({\mathcal {M}})\), then completing in the weak topology would give rise to functions that cannot even be understood as measurable functions on \({\mathcal {M}}\) because of their values ‘at infinity’ (cf. footnote 15).
Even though \({\mathcal {Q}}\) is only required to be linear and not to be a homomorphism (i.e., it is not required to preserve multiplication and thus not required to preserve functional relations in general), it seems natural to expect \({\mathcal {Q}}\) to preserve functional relations on abelian subalgebras of its range. In other words, it seems natural to expect \({\mathcal {Q}}(g(f)) = g({\mathcal {Q}}(f))\) for \(f\in C_0({\mathcal {M}})\) and any Borel function \(g:{\mathbb {R}}\rightarrow {\mathbb {R}}\), as in the usual functional calculus.
This is meant only for the case of finitely many degrees of freedom on a simply connected phase space. It is well known that for systems with infinitely many degrees of freedom or with superselection rules, one requires a different algebra.
Landsman (1998, 2006) also discusses another quantization map that he calls Weyl quantization. This map does not have the Weyl algebra as its image, and so it should be distinguished from the quantization map defined above from \({\mathcal {W}}_C\) to \({\mathcal {W}}\). It is worth reiterating that what is at issue in this paper is the quantum algebra that is the image of a quantization map, but we have not discussed the details of the different mappings one might use once we have fixed some quantum algebra.
I thank an anonymous referee for this point.
References
Ashtekar, A. (2009). Some surprising implications of background independence in canonical quantum gravity. General Relativity and Gravitation, 41, 1927–1943. doi:10.1007/s10714-009-0802-1.
Ashtekar, A., & Isham, C. (1992). Inequivalent observable algebras. Another ambiguity in field quantisation. Physics Letters B, 274, 393–398.
Bratteli, O., & Robinson, D. (1987). Operator algebras and quantum statistical mechanics. New York: Springer.
Clifton, R., & Halvorson, H. (2001). Are Rindler quanta real? Inequivalent particle concepts in quantum field theory. British Journal for the Philosophy of Science, 52, 417–470.
Corichi, A., Vukašinac, T., & Zapata, J. (2007). Polymer quantum mechanics and its continuum limit. Physical Review D, 76, 044016-1-16. doi:10.1103/PhysRevD.76.044016.
Earman, J., & Fraser, D. (2006). Haag’s theorem and its implications for the foundations of quantum field theory. Erkenntnis, 64, 305344. doi:10.1007/s10670-005-5814-y.
Emch, G. (1972). Algebraic methods in statistical mechanics and quantum field theory. New York: Wiley.
Emch, G. (1997). Beyond irreducibility and back. Reports on Mathematical Physics, 40(2), 187–193.
Feintzeig, B. (2016). Toward an understanding of parochial observables. British Journal for the Philosophy of Science. doi:10.1093/bjps/axw010.
Fell, G., & Doran, R. (1988). Representations of *-algebras, locally compact groups, and Banach *-algebraic bundles. Boston: Academic Press.
Haag, R. (1992). Local quantum physics. Berlin: Springer.
Halvorson, H. (2004). Complementarity of representations in quantum mechanics. Studies in the History and Philosophy of Modern Physics, 35, 4556. doi:10.1016/j.shpsb.2003.01.001.
Halvorson, H. (2006). Algebraic quantum field theory. In J. Butterfield & J. Earman (Eds.), Handbook of the philosophy of physics (pp. 731–864). New York: North Holland.
Kadison, R., & Ringrose, J. (1997). Fundamentals of the theory of operator algebras. Providence, RI: American Mathematical Society.
Landsman, N. P. (1998). Mathematical topics between classical and quantum mechanics. New York: Springer.
Landsman, N. P. (2006). Between classical and quantum. In J. Butterfield & J. Earman (Eds.), Handbook of the philosophy of physics (pp. 417–553). New York: North Holland.
Manuceau, J., Sirugue, M., Testard, D., & Verbeure, A. (1974). The smallest C-algebra for the canonical commutation relations. Communications in Mathematical Physics, 32, 231–243.
Petz, D. (1990). An invitation to the algebra of canonical commutation relations. Leuven: Leuven University Press.
Reed, M., & Simon, B. (1980). Functional analysis. New York: Academic Press.
Ruetsche, L. (2003). A matter of degree: Putting unitary inequivalence to work. Philosophy of Science, 70(5), 1329–1342.
Ruetsche, L. (2011). Interpreting quantum theories. New York: Oxford University Press.
Sakai, S. (1971). C*-algebras and W*-algebras. New York: Springer.
Simonnet, M. (1996). Measures and probabilities. New York: Springer.
Acknowledgements
I would like to thank Sam Fletcher, Zacharias Flouris, Hans Halvorson, John Manchak, Sarita Rosenstock, Jim Weatherall, and an anonymous referee for the many helpful comments and conversations that led to this paper. This material is based upon work supported by the National Science Foundation Graduate Research Fellowship under NSF Grant No. DGE-1321846. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.
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Feintzeig, B.H. On Theory Construction in Physics: Continuity from Classical to Quantum. Erkenn 82, 1195–1210 (2017). https://doi.org/10.1007/s10670-016-9865-z
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DOI: https://doi.org/10.1007/s10670-016-9865-z