On Feynman's View of

ANTIMATTER

Bryan W. Roberts, LSE

These Slides: personal.lse.ac.uk/robert49/talks/vienna
Join: philosophyofphysics.org

"I did not take the idea that all the electrons were the same one from him as seriously as I took the observation that positrons could simply be represented as electrons going from the future to the past in a back section of their world lines. That, I stole!" (Feynman, 1972 Nobel Lecture)

The Feynman View of Antimatter:

• Future-directed antimatter is past-directed matter.
• Future-directed matter is past-directed antimatter.

Problem: It is strictly false.

Physical process

Reverse physical process

Not antimatter

Not antimatter

Interpreting Feynman's View of Antimatter

1. Feynman diagrams

2. CT and CPT Invariance

3. Quantization

4. Group Representation

Interpreting Feynman's View of Antimatter

1. Feynman diagrams

2. CT and CPT Invariance

3. Quantization

4. Group Representation

Interpreting Feynman's View of Antimatter

1. Feynman diagrams

2. CT and CPT Invariance

3. Quantization

4. Group Representation

Interpreting Feynman's View of Antimatter

1. Feynman diagrams

2. CT and CPT Invariance

3. Quantization

4. Group Representation

1. Feynman diagrams

1. Feynman diagrams

1. Feynman diagrams

1. Feynman diagrams

1. Feynman diagrams

Reversing a Feynman diagram exchanges matter-antimatter and time

1. Feynman diagrams

Problems:

• A Feynman diagram is not a spacetime diagram.
• A Feynman diagram is not a description of matter but a term in a perturbation series approximation.

1. Feynman diagrams

\( \langle\psi_a,U_t\psi_b\rangle \approx \langle\psi_a,U_t^{(0)}\psi_b\rangle + \langle\psi_a,U_t^{(1)}\psi_b\rangle + \langle\psi_a, U_t^{(2)}\psi_b\rangle + \cdots \)

Cf. BWR and Butterfield (2020)

Interpreting Feynman's View of Antimatter

1. Feynman diagrams

2. CT and CPT Invariance

3. Quantization

4. Group Representation

2. CT and CPT Invariance

2. CT and CPT Invariance

• Classical fields are generally CT invariant.
• Quantum fields are generally CPT invariant.

2. CT and CPT Invariance

Reference frame invariance

2. CT and CPT Invariance

Reference frame invariance

2. CT and CPT Invariance

Parity invariance

Violated by Cobalt-60 decay (Wu et al. 1957)

2. CT and CPT Invariance

Time reversal invariance

Violated by \(K^0\) decay (Cronin and Fitch 1964)

2. CT and CPT Invariance

CPT invariance

• Satisfied in all known interactions
• CPT theorem: true for 'reasonable' Lorentz-invariant fields
  (Lüders 1954, Pauli 1955, Jost 1957)

2. CT and CPT Invariance

CPT invariance

Proposal: time reversal 'really means' \(CT\) or \(CPT\)

"deserves" = ethics? sociology? linguistics?

2. CT and CPT Invariance

Problems:

• Physics, not ethics, sociology, linguistics
• Obfuscates the meaning of \(C\), \(P\), and \(T\).

2. CT and CPT Invariance

2. CT and CPT Invariance

2. CT and CPT Invariance

2. CT and CPT Invariance

Theorems that establish the meaning of \(T\):

• \(T\) is antiunitary if it reverses time translations.
• \(T\) is unique in an irrep of the CCRs or the CARs.
• \(P\) is unique in an irrep of the CCRs or the CARs.

Cf. BWR (2017, 2022)

Interpreting Feynman's View of Antimatter

1. Feynman diagrams

2. CT and CPT Invariance

3. Quantization

4. Group Representation

3. Quantization

'Quantizing \(T\) produces \(CT\)'

3. Quantization

\( \{q,p\} = 1 \) \([q,p]=i\hbar\)

\((q,p) = \) phase space coordinate

3. Quantization

\( \{q,p\} = 1 \) \([Q,P]=i\hbar\)

\((q,p) = \) phase space coordinate

\(Q,P = \) self-adjoint operators

3. Quantization

'Canonical' ('Dirac') Quantization

\(q\mapsto Q, \;\; p\mapsto P, \;\; r\mapsto R, \dots\)

3. Quantization

"[S]tart from a classical field theory, with assumptions about which classical transformations deserve the names 'time reversal' and 'parity reversal' already in place (never mind whence!); obtain a QFT by quantization; work out which transformations on QFT states and operators are induced by the already-named transformations on classical fields, and name the former accordingly.... when one carries out this latter project, with standard names for the classical transformations, the transformation that is usually called 'TC' receives the name 'T'."

— Greaves (2010 BJPS, p.39)

Also: Greaves (2008), Arntzenius and Greaves (2009)

3. Quantization

• Classical field theory: \((S,\Omega,\varphi_t)\) with time reversal \(T_\text{(classical)}\)
• Quantize to get a QFT: \((\mathcal{H},\mathcal{A},U_t)\) with time reversal \(T\)
• Question: \(T_\text{(classical)} \mapsto T\)   or   \(T_\text{(classical)}\mapsto CT\)?
• \(T_{(classical)}\) conjugates complex fields... so maybe \(T_{(classical)} \mapsto CT\)?
• No.

3. Quantization

• Conflating different notions of 'complex conjugation' (Wallace 2009, Baker and Halvorson 2010)
  • Complex classical field: \(\exp(i\omega t)\mapsto \exp(-i\omega t)\)
  • Hilbert space complex structure: \(T:\mathcal{H}\rightarrow\mathcal{H}\)
• Opposite wave-vector of antimatter means the wave-vector is past-directed, but four-momentum is still future-directed.
• 'Canonical Quantization' is not well-defined: the Groenewald-van Hove theorem implies it is impossible.
• Segal quantisation: When you follow quantisation through carefully, \(T_{(classical)} \mapsto T\). (BWR 2022, §8.2.2)

Interpreting Feynman's View of Antimatter

1. Feynman diagrams

2. CT and CPT Invariance

3. Quantization

4. Group Representation

4. Group Representation

Matter and antimatter are related by 'time-reversed representations'

4. Group Representation

Representation view: Symmetries in state space acquire meaning from a group representation. (Wigner 1939, BWR 2022)

4. Group Representation

Meaning of rotation

4. Group Representation

Meaning of parity?

4. Group Representation

• Parity is the only non-trivial symmetry of the rotation group

4. Group Representation

• Parity is the only non-trivial symmetry of the rotation group

4. Group Representation

• Parity is the only non-trivial symmetry of the rotation group

4. Group Representation

• Parity is the only non-trivial symmetry of the rotation group

4. Group Representation

• Parity is the only non-trivial symmetry of the rotation group

A 'flip' is an automorphism of rotations.

4. Group Representation

• Parity is the only non-trivial symmetry of the rotation group
• Time reversal is the only non-trivial symmetry of time translations.
• \((\iota,p,\tau,p\tau)\) are the non-trivial symmetries of the continuous Lorentz group (of boosts, rotations, translations).
• Charge conjugation \(C\) is a symmetry of a gauge group

4. Group Representation

• Wigner (1939): Hilbert space irreps of the Poincaré group determine the possible mass and spin.
• DHR (1971) Superselection Theory: Representations of a (compact) gauge group.
• Matter-Antimatter relationship: Matter and antimatter are described by `conjugate' representations related by an antiunitary
• Antiunitary transformations are time-reversing
  (every antiunitary \(J\) can be written \(J = UT\) for some \(U\)).