The Philosophy of Quantum Field Theory

Abstract

If we divide our physical theories (somewhat artificially) into theories of matter and theories of spacetime, quantum field theory (QFT) is our most fundamental empirically successful theory of matter. As such, it has attracted increasing attention from philosophers over the past two decades, beginning to eclipse its predecessor theory of quantum mechanics (QM) in the philosophical literature. Here I survey some central philosophical puzzles about the theory's foundations. Inequivalent representationsPerhaps the deepest new conceptual problem prompted by QFT, and certainly the one most thoroughly explored by philosophers, is the problem of inequivalent representations.1 From ordinary QM we are familiar with the use of an algebra of operators to stand for the canonical observable quantities (position and momentum, or their field-theoretic equivalents 2 ). The minimal operator algebra for these canonical observables, called the algebra of observables A, can be represented by operators on a Hilbert space H. This is done by assigning an operator π(A) from the bounded operators on H to every operator A ∈ A. The mapping π is called a Hilbert space representation of A. Given plausible assumptions, there is a unique 1 To my knowledge, Huggett and Weingard (1996) were the first to discuss this problem in a published article, although Arageorgis (1995) had already explored it in some depth in his seminal dissertation.2 In bosonic field theories, the canonical observables are the field operators φ(x) and their conjugate momenta π(x), while in fermionic theories only even polynomials of the fields and their momenta are observables.

How to Be a Relativistic Spacetime State Realist

“…One prominent objection to SSR is that we do not adequately understand the relevant instrumental connections (2). Both Lewis (2013) and Baker (2016) worry that the view makes the relationship between fundamental quantities and experimental predictions hopelessly obscure, if not incoherent. Lewis observes that density operators are represented by complex matrices, "but the matrix elements themselves determine a probability distribution over outcomes.…”

“…There may be other viable strategies as well. Baker (2016) is right to point out that "a perspicuous explanation for the expectation values is a major missing piece in extant field interpretations" like SSR. But by recasting the quantum state as a section of a highly structured presheaf, we can start to see how such an explanation might be possible, as well as some of the broad contours it could take.…”

The Coalescence Approach to Inequivalent Representation: Pre-QM∞ Parallels

“…However, many of these observables are physically irrelevant to characterize the actual state of a system, and so Universalism seems to fail O-soundness. Baker [2016] expresses (but does not endorse!) this point: Consider any two inequivalent, irreducible subrepresentations of the large representation.…”

“…See reviews byFriederich [2013];Wallace [2014].2 Lupher [2008] andBaker [2016] are the only notable exceptions that I know of.3 For example,Kronz and Lupher [2005] defend so-called 'Universalism', of whichBaker [2009Baker [ , 2016 favours a 'tempered' version Feintzeig [2016Feintzeig [ , 2018a. offers various arguments in support of 'Algebraic Imperialism'.…”

“…I will not discuss classical analogues of Algebraic Imperialism Feintzeig [2016]. constructs an algebraic version of classical field theory, but this construction is irrelevant for our purposes as the focus of this subsection is on particle theories with varying N .12Baker [2016] objects that this argument simply 'amounts to the observation that, on a universalist-style view, physically contingent facts have a significant role in determining which physical quantities [...] provide physically significant information'. But this misses the point, which is exactly that Universalism, unlike the CA, is not sensitive to the way in which contingent facts determine which observables are most significant.13 An anonymous reviewer has suggested that the pristine ideal can model physical practice in terms of idealizations.…”