The fate of ‘particles’ in quantum field theories with interactions Doreen Fraser Department of Philosophy University of Waterloo

Context quanta: entities that are particle-like insofar as they are countable and have appropriate energies (i.e., in QFT, the same energies as classical relativistic particles); these entities do not bear labels (Redhead, French, Teller) Other Results: 1. Hagerfeldt-Malament-Halvorson-Clifton no-go theorems for free systems: quanta are not localizable 2. Unruh effect for free systems and accelerating observers: different observers adopt different quanta representations My Conclusion: QFT does not describe quanta because interacting systems cannot be given a quanta interpretation DF, “The fate of ‘particles’ in quantum field theories with interactions,” Studies in History and Philosophy of Modern Physics 39 (2008)

Sketch of the argument Three failed attempts to give interacting systems a quanta interpretation Attempt #1: Use the Fock space representation for a free system to represent the interacting system → ruled out by Haag’s theorem (1955) Attempt #2: Quantize analogously to the free case; i.e., Fourier decompose the interacting field into positive and negative frequency parts to obtain creation, annihilation, and particle number operators → the operators are not Poincaré covariant

Sketch of the argument Three failed attempts to give interacting systems a quanta interpretation Attempt #3: Define annihilation, creation and number operators in formal analogy to the free case: → |0> is not the vacuum state and c † (k,t)|0> cannot be interpreted as a one quantum state because it does not possess the relativistic energy

Diagnosis Why do attempts to give a quanta interpretation for interacting systems fail? Not (merely) a problem created by the n ≥ 2 states Not a problem created by the quantum theoretic assumptions of QFT Special relativity is the culprit Attempt #1: Haag’s theorem relies on relativistic assumptions Attempt #2: The Fourier decomposition is not covariant under Poincaré transformations Attempt #3: The purported “one-quantum” state does not satisfy the relativistic energy constraint

Asymptotic quanta interpretation for interacting systems? At asymptotic times, interactions are negligible. Can a quanta interpretation for an interacting system be based on the Fock representation for a free system at asymptotic times? No. The pertinent question is whether the formalism describes entities which possess particle-like properties at intervening times.

Conclusions QFT does not describe quanta because interacting systems cannot be given a quanta interpretation i.e., interacting systems do not even admit an interpretation in terms of quanta that are not countable and not localizable it is not just the n ≥ 2 states that are problematic it is the special relativistic—and not the quantum theoretic—ingredients of QFT that are to blame