Are anti-particles particles traveling back in time?

- Classical preliminariesHilary Greaves & Frank Arntzenius

TAU workshop

June 12, 2006

Feynman on time reversal in QFT “A backwards-moving electron when viewed with

time moving forwards appears the same as an ordinary electron, except it’s attracted to normal electrons – we say it has positive charge. For this reason it’s 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)

Questions

(1) What is the right time reversal operation? (2) What is the methodology for answering

(1)? (3) [Why] should we care about (1)?

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

1.1 A simple electromagnetic world E, B fields on each timeslice Worldline of a charged particle Maxwell’s equations:

Lorentz force law:

E

B

EB

t

BdE

dt

0

0

q E V B ma

EB

BEt

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

EB

t

EB

BEt

EB

BE

t

q E V B ma

1.3 Albert’s 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 S1, …, SF which is in accord with the dynamical laws of this theory, SF, …, S1 is too. And so this theory is not invariant under time reversal. Period.” (Albert (2000), p.14)

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

2.1 Malament’s 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

M g W F, , , , ( )

v a

v a

m q,

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,

F abv a

v a

m q,

a

F

Electro-magnetic field

Representative of electromagnetic field

F ab

2.3 Time reversal

We have:

represented by ,

where va, Fab are defined from W, F (resp.) relative to a choice of temporal orientation

Time reversal: is just

and hence

M g W F, , , , ( ) M g v Fa ab, , , ,

M g W F M g W Fa a, , , , ( ) , , , , ( )

M g v F M g v Fa ab a a ab a, , , , , , , ,

2.4 Time reversal invariance

Maxwell’s equations & Lorentz force law:

Time reversal:

Equations invariant under time reversal

[ ]a bc

aab

ab

b aa

b

F

F

qF v m v v

0

0

F F

v v

ab ab

a a

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

Magnetic field:

E Fa ab

b:

B Fa abcdb cd:

1

2

a

( )( ) F Eab

b a

1

2( )( )( ) abcd

b cdaF B

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

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)

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 we’re willing to be sufficiently liberal about finding ‘the time reversal operation’.

3.3 Toy model: a simple theory that is NOT!! time reversal invariant

r

Dynamical law:

dr = -krdtGeneral solution:r(t)=e-kt+δ

3.4 ‘Time reversal operation’ for our toy theory

r1

r

Time reverse of a state r:

rr

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 isn’t a game worth playing The ‘pragmatic program’ – weak constraints The ‘ontological program’ – strong constraints

3.6 Response 1: It isn’t a game worth playing “Time reversal, schmime reversal”

What’s in a name??

E EB -B

d/dt -d/dt

invariant

E EB B

d/dt -d/dt

not invariant

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’.

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

Albert’s ‘partial time reversal invariance’

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

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

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

M , ,

M M, , , , ?

3.12 Time reversal for the ontological program If that’s our interest, then the time reversal

operation should (obviously!) be:

… and whatever follows from this…

M M, , , ,

3.13 Q: Is Malament’s the only way?

“The inversion of magnetic fields [under time reversal] is, in fact, forced by elementary geometric considerations.”

Malament (2003)

3.14 ‘Albert’s’ ontology for classical EM Suppose that the E and B fields are fundamental (!). (Fab 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 Ockham’s Razor favors Malament’s ontology over Albert’s

(twice)

M g E B v S M g E B v Sa a a a a a, , , , , , , , , , , ,

M g F v S M g F v Sab a ab a, , , , , , , ' , , ,

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

4.1 The Third Way: ‘Feynman’s proposal’ Back to (unadulterated) Minkowski spacetime Key idea: suppose that worldlines of charged particles are

intrinsically directed. [cf. Feynman diagrams]

v a

v a

m q,

v a

m q,

( )

4.2 ‘Feynman’s’ 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, Fab

So: Fab is defined independently of temporal orientation

4.3 Time reversal on ‘Feynman’s’ view

M g v F M g v Fa ab a a ab a, , , , , , , ,

v a

m q,

v a

m q,

4.4 E and B fields, on the ‘Feynman’ proposal Define these from Fab, just as before:

These are not the textbook transformations for E and B

But the theory is still time reversal invariant

E Fa ab

b: ( )( )F Eab

b a

B Fa abcdb cd:

1

2

1

2( )( )( ) abcd

b cdaF B

4.5 Time reversal invariance of classical EM‘Feynman’

[ ]a bc

aab

ab

b aa

b

F

F

qF v m v v

0

0

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

5.1 Playing baseball, Malament style Harry Mary

Harry Mary

Harry Mary

5.2 Playing baseball, Feynman style Harry Mary Harry Mary

Harry Mary

5.3 Distinguishing particles and antiparticles (Malament)

Lorentz force law:

v av a

m q, m q,

qF v m v vab

b aa

b

Particle Antiparticle

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)

qF v m v vab

b aa

b

v a

m q,

v a m q,

v a

m q,

v am q,

P P

A A

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

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

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…