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Are anti-particles particles traveling back in time? - Classical preliminaries Hilary Greaves & Frank Arntzenius TAU workshop June 12, 2006

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Feynman on time reversal in QFT A backwards-moving electron when viewed with time moving forwards appears the same as an ordinary electron, except its attracted to normal electrons – we say it has positive charge. For this reason its called a positron. The positron is a sister to the electron, and it is an example of an anti- particle. This phenomenon is quite general. Every particle in Nature has an amplitude to move backwards in time, and therefore has an anti- particle. (Feynman, QED, p.98)

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Questions (1) What is the right time reversal operation? (2) What is the methodology for answering (1)? (3) [Why] should we care about (1)?

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Outline of the talk (1) Albert on time reversal in classical EM (2) Malament on time reversal in classical EM (3) On methodology (4) Feynman on time reversal in classical EM (5) Baseballs and antiparticles (6) The story so far

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1.1 A simple electromagnetic world E, B fields on each timeslice Worldline of a charged particle Maxwells equations: Lorentz force law: E B BE t

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1.2 What is the time reverse of this world? The textbooks: E E, B -B Theory time reversal invariant David Albert: E E, B B Theory not time reversal invariant E B BE t E B BE t

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1.3 Alberts argument What counts as an instantaneous state of the world, according to classical electrodynamics… is a specification of the positions of all the particles and of the magnitudes and directions of the electric and magnetic fields… And it turns out not to be the case that for any sequence of such states S 1, …, S F which is in accord with the dynamical laws of this theory, S F, …, S 1 is too. And so this theory is not invariant under time reversal. Period. (Albert (2000), p.14)

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Outline of the talk (1) Albert on time reversal in classical EM (2) Malament on time reversal in classical EM (3) On methodology (4) Feynman on time reversal in classical EM (5) Baseballs and antiparticles (6) The story so far

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2.1 Malaments account of time reversal in classical electromagnetism Fundamental quantities: EM field, worldlines of charged particles How to represent a worldline: future- directed unit tangent at each point

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2.2 Representing the world in the language of differential geometry: The EM field EM field, F: a map from tangent lines to (acceleration) 4-vectors Map from 4-vectors to 4-vectors: rank 2 tensor field Representative of electromagnetic field, Electro- magnetic field Representative of electromagnetic field

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2.3 Time reversal We have: represented by, where v a, F ab are defined from W, F (resp.) relative to a choice of temporal orientation Time reversal: is just and hence

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2.4 Time reversal invariance Maxwells equations & Lorentz force law: Time reversal: Equations invariant under time reversal

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2.5 You want to know about the E and B fields? OK, OK… Frame: future-directed timelike vector field, Electric field: Magnetic field:

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Outline of the talk (1) Albert on time reversal in classical EM (2) Malament on time reversal in classical EM (3) On methodology (4) Feynman on time reversal in classical EM (5) Baseballs and antiparticles (6) The story so far

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3.1 How to beg the question Begging the question: Postulate that the theory is time reversal invariant, and derive the time reversal operator from that assumption E.g.: Finding the time reversal operator for a quantum field theory: …impose the constraint that time reversal should be a symmetry of the free Dirac theory, [T,H]=0… (Peskin & Schroeder 1995, p.67)

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3.2 The triviality problem Claim: There are theories that obviously ought not to count as time reversal invariant, but that will count as time reversal invariant if were willing to be sufficiently liberal about finding the time reversal operation.

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3.3 Toy model: a simple theory that is NOT!! time reversal invariant r Dynamical law: dr = -kr dt General solution: r(t)=e -kt+δ

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3.4 Time reversal operation for our toy theory Time reverse of a state r: r r

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3.5 Three responses to the triviality problem To assume that the theory must be time reversal invariant, and derive the time reversal operation from that assumption, is question-begging. If talking about time reversal is to be a game worth playing, there must be some constraints on the time reversal operation. Three responses: Fine – it isnt a game worth playing The pragmatic program – weak constraints The ontological program – strong constraints

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3.6 Response 1: It isnt a game worth playing Time reversal, schmime reversal Whats in a name?? E B -B d/dt -d/dt invariant E B d/dt -d/dt not invariant

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3.7 Response 2: The pragmatic program Any transformation such that the transformation looks something like time reversal in some sense, and the theory is invariant under that transformation, is worth calling time reversal.

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3.8 Pragmatism (I): Macroscopic time- asymmetry Why do ripples on a pond always spread outwards? Why do eggs always break? Why does steam always rise? This is puzzling if the microscopic theory is time- reversal invariant in any reasonably intuitive sense Alberts partial time reversal invariance

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3.9 Pragmatism (II): finding conserved quantities The point is that any operator that commutes with the S-matrix is valuable. We regard the words [time reversal] as merely suggesting a particularly fertile area in which such operators might be found. Geroch 1973, p.104

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3.10 Response 3: The ontological program The time reversal invariance (or otherwise) of our best theories gives us a clue about the structure of spacetime: is there a preferred temporal orientation? Physics is time/space translation invariant infer that there is no preferred location Physics is Lorentz invariant infer that there is no preferred foliation If physics is time reversal invariant, infer that there is no preferred temporal orientation

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3.11 Time reversal for the ontological program The aim: define a notion of time reversal operation that is suited to the pursuit of this program In coordinate-free language, our question is: is the class of models invariant under the transformation

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3.12 Time reversal for the ontological program If thats our interest, then the time reversal operation should (obviously!) be: … and whatever follows from this…

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3.13 Q: Is Malaments the only way? The inversion of magnetic fields [under time reversal] is, in fact, forced by elementary geometric considerations. Malament (2003)

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3.14 Alberts ontology for classical EM Suppose that the E and B fields are fundamental (!). (F ab is a mere construct.) Then, time reversal will not flip the sign of the B field. What follows from this: The theory is not Lorentz invariant – we require a preferred frame The theory is not time reversal invariant – we require a preferred temporal orientation Ockhams Razor favors Malaments ontology over Alberts (twice)

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Outline of the talk (1) Albert on time reversal in classical EM (2) Malament on time reversal in classical EM (3) On methodology (4) Feynman on time reversal in classical EM (5) Baseballs and antiparticles (6) The story so far

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4.1 The Third Way: Feynmans proposal Back to (unadulterated) Minkowski spacetime Key idea: suppose that worldlines of charged particles are intrinsically directed. [cf. Feynman diagrams]

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4.2 Feynmans proposal Then: the EM field is fundamentally a map from 4-vectors to 4-vectors Or just: the EM field is fundamentally a rank 2 tensor field, F ab So: F ab is defined independently of temporal orientation

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4.3 Time reversal on Feynmans view

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4.4 E and B fields, on the Feynman proposal Define these from F ab, just as before: These are not the textbook transformations for E and B But the theory is still time reversal invariant

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4.5 Time reversal invariance of classical EM Feynman

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Outline of the talk (1) Albert on time reversal in classical EM (2) Malament on time reversal in classical EM (3) On methodology (4) Feynman on time reversal in classical EM (5) Baseballs and antiparticles (6) The story so far

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5.1 Playing baseball, Malament style Harry Mary Harry Mary Harry Mary

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5.2 Playing baseball, Feynman style Harry Mary Harry MaryHarryMary

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5.3 Distinguishing particles and antiparticles (Malament) Lorentz force law: ParticleAntiparticle

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A particle and its antiparticle partner have the same charge as, but travel in opposite temporal directions to, one another Particle: a particle traveling forwards in time Antiparticle: a particle traveling backwards in time Then: time reversal turns particles into antiparticles! 5.4 Distinguishing particles and antiparticles (Feynman) PP A A

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Outline of the talk (1) Albert on time reversal in classical EM (2) Malament on time reversal in classical EM (3) On methodology (4) Feynman on time reversal in classical EM (5) Particles and antiparticles (6) The story so far

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6.1 The story so far Ontological time reversal: Time reversal leaves all fundamental quantities, with the (possible) exception of the temporal orientation, invariant (1) What is the right time reversal operation? (A): Depends on which quantities are fundamental (2) What is the methodology for answering (1)? To justify a time reversal operation on a non-fundamental quantity is to explain how its definition in terms of fundamental quantities depends on the temporal orientation, and to derive its time reversal transformation from that definition The Feynman ontology may be preferable to the Malament one

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6.2 The story so far (3) [Why] should we care? [If] the project gives any guidance on which quantities in a theory should be regarded as fundamental [If] the project helps illuminate the relationship of our theories to spacetime structure [If] the project illuminates anything in QFT…

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