Philosophical Aspects of Quantum Field Theory: II
چکیده
According to a regnant criterion of physical equivalence for quantum theories, a quantum field theory (QFT) typically admits continuously many physically inequivalent realizations. This, the second of a two-part introduction to topics in the philosophy of QFT, continues the investigation of this alarming circumstance. It begins with a brief catalog of quantum field theoretic examples of this non-uniqueness, then presents the basics of the algebraic approach to quantum theories, which discloses a structure common even to ‘physically inequivalent’ realizations of a QFT. Finally, it introduces and evaluates a handful of strategies for interpreting quantum theories in the face of the non-uniqueness of their Hilbert space representations. Part I of this mathematically informal introduction to interpretive questions that arise in connection with quantum field theory (QFT) highlighted a signal disanalogy between ordinary non-relativistic quantum mechanics (QM) and what it called QM¥. Ordinary QM lies within the scope of the Stone–von Neumann and Wigner–Jordan theorems, which guarantee the uniqueness of the representation of the canonical commutation relations (CCRs) (Stone von-Neumann) or canonical anticommutation relations (CARs) (Wigner–Jordan) constituting the quantum theory of a system with finitely many degrees of freedom. For n finite, the Jordan–Wigner theorem assures us, any representation of the CARs for n spin systems is unitarily equivalent to any other. Mutatis mutandis (and with a few qualifications (cf. Summers 2001), the Stone–von Neumann theorem assures us that representations of the CCRs are unique up to unitary equivalence. As unitary equivalence is both a plausible and a venerable criterion for the physical equivalence of quantum theories, these theorems are taken to establish that the physical theory obtained by carrying out the instruction ‘quantize this system’ is unique, provided the system possesses only finitely many degrees of freedom. Theories of QM¥ concern systems possessing infinitely many degrees of freedom: QFTs, which associate a field amplitude and a conjugate momentum with every point of spacetime, and the thermodynamic limit of statistical mechanics, reached by letting the number of microsystems considered and the volume they occupy go to infinity, while keeping their density finite. Theories of QM¥ fall outside the scope of the Stone–von Neumann and Jordan–Wigner theorems. They admit presumptively physically inequivalent realizations. This essay considers interpretive responses to that circumstance. Section 1 documents examples of unitarily inequivalent quantizations of the simplest imaginable classical field theory, the Klein–Gordon theory, and considers repercussions of this non-uniqueness for the idea that QFT – understood by its practitioners as particle physics – sustains a particle interpretation. The consideration is continued in Section 2’s account of Haag’s theorem and interacting QFTs. Section 3 sketches the rudiments of the algebraic approach to quantum theories, an approach which enables us not only to Philosophy Compass 7/8 (2012): 571–584, 10.1111/j.1747-9991.2011.00387.x a 2012 The Author Philosophy Compass a 2012 Blackwell Publishing Ltd compare different unitarily inequivalent realizations of a quantum theory, but also to see what those realizations might have in common. Section 4 uses the algebraic apparatus to articulate, and begin to evaluate, a variety of approaches to interpreting QM¥. 1. The Free Klein–Gordon Field: Particle Ontology The theory of the mass m free boson field, obtained by quantizing the classical mass m Klein–Gordon field in Minkowski spacetime, may be the simplest QFT there is. The aim of section’s impressionistic treatment is to place this theory of QM¥ outside the scope of the Stone–von Neumann theorem. First, some spacetime preliminaries (for a proper introduction, consult Wald 1984) Minkowski spacetime, the spacetime of Special Relativity, is highly symmetric. It admits a (indeed, many) coordinate system such that the light cone structure at each point assumes the familiar 45 shape in that coordinate system. Less pictorially put, the metric of Minkowski spacetime is unchanged under a group of transformations that include translations, rotations, and Lorentz boosts (which can be thought of as translating between inertial coordinate systems adapted to observers moving at constant velocity with respect to one another). These symmetries of the metric, aka isometries, form a group called the Poincaré group. General Relativity traffics in more general spacetimes, typically with a much more variable light cone structure. An arbitrary general relativistic spacetime – a solution to Einstein’s field equations – needn’t have any isometries whatsoever. Where (x1, x2, x3, t) are inertial coordinates for Minkowski spacetime, the classical Klein–Gordon equation for mass m is @ @x1 þ @ 2 @x2 þ @ 2 @x3 @ 2 @t2 m uðxÞ 1⁄4 0: ð1Þ Solutions /(x) to (1) are maps from points of Minkowski spacetime to C. To follow the Hamiltonian recipe for quantizing classical Klein–Gordon theory is to find a Hilbert space representation of CCRs involving the field amplitudes u(x) and conjugate momenta p(x) (see Wald (1994, §3.1) for details). In Minkowski space-time, such a representation can be obtained by following a textbook ‘frequency-splitting’ heuristic for quantizing Klein–Gordon theory (for which, see Kaku 1993, Ch. 3.2; Wald 1994, Ch. 4.3). Solutions to the Klein–Gordon equation can be Fourier-decomposed into uncoupled normal modes with angular frequency xk. This suggests an infinite collection of independent oscillators as a model of the classical field. The standard quantization of the simple harmonic oscillator deploys ‘raising’ and ‘lowering’ operators interdefined with p̂ and q̂ and facilitating a perspicuous treatment of system energetics. Exploiting the analogy with the simple harmonic oscillator, the textbook approach represents the CCRs quantizing Klein–Gordon theory by means of ‘raising’ and ‘lowering’ operators â y k and âk for positive frequency (with respect to inertial time t) field modes xk: 1⁄2̂ak; âk0 1⁄4 0 1⁄4 âk; â y k0 h i ; âk; â y k0 h i 1⁄4 dkk0 Î : ð2Þ These operators (which are more commonly, albeit misleadingly (see Teller 1995), known as ‘creation and ‘annihilation’ operators) act on a kind of Hilbert space known as a symmetric Fock space (see Wald 1994, A.2 for a definition). The action of the raising and lowering operators facilitates the interpretation of quantized Klein–Gordon theory as particle physics. The Fock space features a state |0æ such 572 Aspects of QFT II a 2012 The Author Philosophy Compass 7/8 (2012): 571–584, 10.1111/j.1747-9991.2011.00387.x Philosophy Compass a 2012 Blackwell Publishing Ltd that âkj0i 1⁄4 0 for all k. |0æ, called the vacuum state, is the lowest energy eigenstate of the quantum Hamiltonian for the free boson field (also handled by analogy with the simple harmonic oscillator). The state â y kj0i is an eigenstate of the Hamiltonian with the same total energy a particle with frequency xk would have – provided the momentum and rest masses of this particles are given by standard relativistic expressions. The pattern reiterates. The state |nkæ, obtained by acting on the vacuum n times with â y k and normalizing, has the energy appropriate to n particles of frequency xk, and so on. Acting on |nkæ with âk and normalizing yields a state with the energy appropriate to n)1 particles of frequency xk, and so on. (It is important to emphasize that these are truths about the relations between Fock space vectors and raising and lowering operators. While the truths tempt an understanding of the action of the raising operator as an event of physical particle creation, that is an optional interpretive overlay to the mathematical facts.) The operator N̂k 1⁄4 âkâk is known as the number operator for particles of variety k. Its eigenvalues are natural numbers; the state |nkæ is an N̂k eigenstate with eigenvalue n. Thus N̂k is understood to count particles of variety k: its expectation value in a state gives the mean number of particles of variety k one would find in that state. Hence, N̂ 1⁄4 P j â y j âj is the total number operator. That for all k, h0jN̂kj0i 1⁄4 h0jN̂ j0i 1⁄4 0 reinforces |0æ’s claim to be the vacuum or no-particle state. States that admit analysis in terms of their particle contents are not isolated. Every state in the Fock space can be approximated to arbitrary precision by acting on |0æ with polynomial combinations of raising and lowering operators. Still, it must be noted just how radically a particle concept framed by QFT departs from our ingrained particulate expectations. Particles, as we think of them, are confined to finite, bounded regions of spacetime. A number of results (Malament 1996; Halvorson and Clifton 2002) suggest that such confinement cannot be had in QFT. Particles, as we think of them, possess what the scholastics called ‘haecceity’, Teller (1995, 17) ‘primitive thisness,’ and Redhead (1983, 59) ‘transcendental identity.’ An extensive literature (see Castellani (1998) for a sample) questions whether bosons and fermions can exhibit this and yet abide by the non-Maxwell-Boltzmann statistics characteristic of their aggregates. Particles, as we think of them, are persistent in their presence and inert in their absence. Excitations of the quantum field are subject to creation and annihilation; even the vacuum fluctuates (Redhead, op. cit.). Particles, as we think of them, come in sets with determinate cardinality. States of the quantum field need not be eigenstates of the total particle number operator, and thus (on the usual way of thinking about quantum systems) need not be states describing a determinate number of particles (Teller 1995, Ch. 2). These are all grounds for doubting that QFT is a theory of particles as we think of them, particles as well-localized, enduring, individuable gobbets of mass-energy. But they leave intact the possibility of a particle concept fundamental to QFT in something like the way position and momentum are fundamental to classical mechanics: just as a classical state can be completely characterized and distinguished from other classical states by specifying its position and momentum, so too a state of the quantum field can be completely characterized in terms of its particle contents. Because the phase space of Klein–Gordon theory is not finite, the Stone–von Neumann theorem cannot vouch that this representation is unique up to unitary equivalence. Indeed, the CCRs for the Klein–Gordon field admit continuously many inequivalent representations. (See Garding and Wightman (1954) or Arageorgis (1995, Appendix C), for a construction of unitarily inequivalent Fock space representations that deploys a move similar to that used in Section 5.1 of Aspects of QFT, Part I to exhibit unitarily inequivalent models of the infinite spin chain.) One symptom of this unitary inequivalence is that the vacuum state of one of a Aspects of QFT II 573 a 2012 The Author Philosophy Compass 7/8 (2012): 571–584, 10.1111/j.1747-9991.2011.00387.x Philosophy Compass a 2012 Blackwell Publishing Ltd pair of unitarily inequivalent representations can’t be realized as a density operator state on the other representation. There is a consortium of tacit assumptions that tends to frame foundational engagement with physics. I’ll call this consortium ‘extremism’. Extremism supposes that there exist fundamental, non-approximate laws that determine which circumstances are physically possible and which aren’t. It supposes further that it is the task of physics to identify these laws, and identify them with precision: physics aims at truth outright, and nothing so hazy as approximate truth. Extremism accordingly supposes that to interpret a physical theory is to take its laws as these fundamental ones, and so to seek to characterize the unique and determinate set of physical possibilities those laws allow. What falls outside this set simply isn’t physically possible, according to the theory so interpreted; what falls within is simply physically possible; because the laws are fundamental and exact, there is nothing in between. Here I will try to suggest that part of what makes QM¥ interesting is the pressure it puts on these extremist assumptions about the character of physics and the task of interpretation. I will develop my suggestion by, until further notice, operating within the scope of these (now not so tacit) extremist assumptions, and identifying the limitations of interpretations of the sort considered. So assume that theorists wielding inequivalent representations are extremists. Each assumes that the set of states countenanced by his or her representation coincides with the unique set of physically possible states. Then the theorists don’t simply disagree about which state is the ground state of the free boson field. Each denies that the other’s ground state is a state at all. That is, they’re not disagreeing about how to describe worlds they both agree are possible. A translation manual of the sort set up by a unitary map between their theories could resolve (or at least drastically temper) that disagreement. They’re disagreeing about which worlds are possible to begin with. Because the sets of worlds each deems possible are (in a sense that can be made precise) disjoint, theorists wielding unitarily inequivalent representations can’t both be even partially right. Regarded as theories of ordinary QM, unitarily inequivalent representations of the CCRs quantizing classical Klein–Gordon theory are out-and-out rivals. This leaves us with continuum-many starkly distinct theories vying for the title of free boson field – a veritable embarrassment of riches for particle physics. 2. Haag’s Theorem: Particle Ontology So far, the QFTs under discussion have been free field theories. Real particle physics treats interactions, often through the so-called ‘interaction picture’. To first approximation, the interaction picture works as follows. What we want is a scattering amplitude between incoming states |winæ (e.g., the state of an electron-postiron pair) and outgoing states |/outæ (e.g., the state of a pair of muons). The gadget that predicts probabilities for this pair-production process is the S-matrix S, in the sense that