1. The fate of ‘particles’ in quantum field theories with interactions Doreen Fraser Department of Philosophy University of Waterloo dlfraser@uwaterloo.ca

2. 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) 841- 859.

3. 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

4. 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

5. 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

6. 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.

7. 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