The Fate of Particles in Finite-Temperature Quantum Field Theory

(Download an old draft of the paper here)

Question: Do we have a well-defined particle concept in finite-temperature quantum field theory (QFT)? To answer this question, we first need to look at the particle concept in the vacuum sector of QFT. Does vacuum QFT support a (fundamental) particle ontology?

In favour of particle ontology: QFTs get most of their use and empirical confirmation from high-energy particle physics. The paradigmatic experiments in particle physics involve colliding beams of subatomic particles together, and analyzing the resulting products of collision. If QFT does not admit a particle ontology, then it is difficult to make sense of the practice of particle physics.

In favour of field ontology: The framework of QFT is a field-theoretic framework. QFTs are constructed by starting with a theory of classical fields and quantizing. It therefore seems most plausible that the underlying ontology would be a field ontology.

Against particle ontology: There cannot be localizble particles in QFT, in the sense of being definitely contained in any finite region (Malament 1995; Halvorson 2001; Halvorson and Clifton 2002). It is a well-known result that particle number in QFT is not conserved; particles can be created and destroyed. Finally, countability and mass-energy relations do not consistently hold in interacting QFT (Fraser 2008).

Against field ontology: Arguments from Fraser (2008) adapted to the context of wave-functional field ontology by Baker (2009).

Free quanta in QFT: countable, aggregable, and eigenstates of the free Hamiltonian. Makes use of the Fock space representation.

Interactions: As Fraser (2008) argues, this quanta representation for free fields cannot be carried over to interacting fields directly. Haag’s theorem shows that Hilbert spaces for free and interacting particles are unitarily inequivalent. Fraser argues that interacting QFT therefore does not “support the inclusion of particlelike entities as fundamental entities in our ontology” (p. 856). Nevertheless, in practice, physicists use the free Fock space representation for weakly interacting theories in some non-fundamental way to connect the formalism of QFT to the phenomenology of particle physics.

Conclusion: No fundamental particle ontology in vacuum QFT. At best an approximately defined particle concept applicable in some contexts.

Approximate quanta in interacting QFT: One can recover the link to particle physics with a much weaker form of particlelike ontology, where we have a notion of “approximately free, approximately localizable” via the S-matrix and interaction picture. Particles far from the scattering region may be treated as approximately free, where one uses the free quanta concept and Fock space. Setting aside issues with how to understand Haag’s theorem for the interaction picture, we have a highly approximate version of particles to fill the required roll (Earman & Fraser 2006, Miller 2018, Koberinski 2023). These are particle surrogates well-suited to cases where the S-matrix is used to model particle collisions with asymptotically free incoming and outgoing particles.

From vacuum to finite-temperature

Particle number operator: In a thermal state, there is no well-defined particle number operator. No Fock space representation.

No free in and out states: S-matrix is not defined for finite temperature. In and out states are excitations about a thermal background, so no free particles at +/- infinity. (Potential exception for asymptotically free QFTs?)

Is this a problem?

1. The paradigmatic use for particle phenomenology—scattering theory—does not exist for finite-temperature QFT. So there is less pressure on recovering a particlelike ontology in this domain.

2. Some particlelike properties remain. One can determine the mass spectrum for fully interacting theory by looking for poles in the n = 2-point functions, and these give some minimal notion of particlelike states for all contexts, including finite-temperature QFT. One could also resort to detector phenomenology: a particle (at finite temperature) is whatever a particle-detector detects.

Lessons for ontology of quantum field theory

Part of the challenge in spelling out a positive ontology for QFT is the fact that it is a theoretical framework, rather than a theory in its own right. Physical content is sparse at the framework level, allowing for many different applications in different contexts. That flexibility, which is a pragmatic virtue of a framework, also makes defining a uniform ontology challenging. Some systems will have an appropriate particle ontology, others fields, and still others something different. One must specify a more concrete theoretical context to have enough physical content for positive, nonfundamental ontological claims. This fact explains why, while negative ontological claims abound for QFT, positive ones are much harder to articulate. We typically don’t expect a uniform ontology for all classical theories, and QFT should be no different.

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