Representing

The Arrow of Time

Bryan W Roberts | LSE

Slides: personal.lse.ac.uk/robert49/talks/smolenice
Book: arxiv:2212.03489

I shall use the phrase "time's arrow" to express this one-way property of time which has no analogue in space.
It is a singularly interesting property from a philosophical standpoint.

— Arthur Eddington (1928)

Some Arrows You've Worked On

Illusion looms

Question: Is there an arrow of time itself?

Question: Is there an arrow of time itself?

Price's Table (Price 1996, Ch.1)

"Is time anisotropic at all, and how could we tell if it is? What could constitute good grounds for taking it to be so, and do we have such grounds?"

— Price (2011, p.192)

Thesis of This Talk:

There is evidence of an arrow of time itself (T-violation)

...but not where it is usually said to be.

Book: Free Open Access

Plan

1. Time reversal
2. A philosophy of time
3. An arrow of time itself
4. Misdirected arrows

A curiosity

\(\psi_t(x) \mapsto \psi_{-t}(x)\) is not a symmetry of Schrödinger's picture

...but \(\psi_t(x)\mapsto\psi^*_{-t}(x)\) usually is a symmetry.

What does it mean to reverse time?

What does it mean to reverse time?

Two camps

"Since \(\psi(x,0)\) encodes information about the momentum of the particle it must be 'turned around' or non-trivially transformed by the time reversal operation so that the reversed state at \(t = 0\) describes the wave packet propagating in the opposite direction."

—John Earman (2003)

"If one surveys the literature concerning this issue, one finds many arguments that attempt to blur the difference between WRI ['Wigner reversal' invariance] and TRI [time reversal invariance]"

— Callender (2000)

Lesson

Time reversal must be as simple as \(t\mapsto-t\).

...and yet physics seems to require more.

Plan

1. Time reversal
2. A philosophy of time
3. An arrow of time itself
4. Misdirected arrows

The representation view

A philosophy of symmetry for physics.

Some Motivation:

\(t\) = two o'clock

\(t\) = time-translate by two hours

Let \(\mathcal{H}\) be a Hilbert space, or \((M,\omega)\) a symplectic manifold.

When do we have a time translation as opposed to a spatial transation?

• Sociology: When we use the letter \(t\).
• Physics: When we have a representation of time translations.

Representing Time

Representing Time

Time translations represented among State space symmetries

Symmetry

...is often said to constrain which laws of physics are possible:
this is a 'meta-law' view of symmetry.

Standard Interpretation. Noether's (first) Theorem: Every variational symmetry admits a conserved current.

• So symmetries imply conservation laws. But...
• Problem 1: Conservation laws also imply symmetries (Lange 2007)
• Problem 2: Multiple symmetries imply the same conservation laws (Brown)

Time Translation Symmetry: Experimental repeatability

Time Translation Symmetry: Experimental repeatability

Claim. Time translation symmetry implies the laws of motion

Dynamical laws are just representations of time translations

• Quantum Mechanics:
  Time translations: \(G=(\mathbb{R},+)\)
  State symmetries: (anti)unitaries \(U : U^*U=UU^*\)
  Representing Time: Homomorphism \(t \mapsto U_t\).
  
  
• \(\psi(t) := U_t\psi \;\Rightarrow\;\) \(i\tfrac{d}{dt}\psi(t) = H\psi(t)\) with \(U_t=e^{-itH}\)
   

Dynamical laws are just representations of time translations

• Classical Mechanics:
  Time translations: \(G=(\mathbb{R},+)\)
  State symmetries: (anti)symplectics \(\phi : \phi^*\omega=|\omega|\)
  Representing Time: Homomorphism \(t \mapsto \phi_t\).
  
  
• \(X = \tfrac{d}{dt}\phi_t \;\Rightarrow\;\) \(\iota_X\omega = dh\) \( \;\Rightarrow\; \dot{q}=\partial h/\partial p, \dot{p} = -\partial h/\partial q \)
  (in each local neighbourhood)

Generalisations of the Representation view

• Poincaré group unitary rep \(P^\uparrow_+ \; \overset{\mbox{\(\phi\)}}{\longrightarrow}\; \phi(P^\uparrow_+)\) (Wigner 1939)
• Diffeomorphism group of timelike translations (Brunetti, Fredenhagen and Verch 2003)
• A discrete group for discrete time, etc.

The representation view

A transformation of a state space can be interpreted as a 'spacetime symmetry' only if it is an element of a representation of a spacetime structure.

BWR (2022, Chapter 2)

Example 1. \(U_t\) means 'time translation' only if it represents a time translation.

Example 2. \(T\) means 'time reversal' only if it represents temporal reflection \(t\mapsto-t\).

A: Yes, including if \(t\) is a time translation.
\(T\) is just the operator that represents it.

Not just coordinates...

...but also relational structure...

...and the representation view.

Question. \(t\mapsto-t\) is a group automorphism , not a group element. So, how can we represent it?

Answer: the group can be extended to include it.

Any group \(S\) of automorphisms of \(G\) defines an extension of the group to \(G\ltimes S\) via the semidirect product.

• \((\mathbb{R},+)\) extends to includetime reversal \(\tilde{G} = (\mathbb{R},+)\ltimes\{I,\tau\}\)
  
  
• \(P^\uparrow_+\) extends to the 'full' Poincaré group \(P = P^\uparrow_+\ltimes\{I,\tau,\pi,\pi\tau\}\)
  
  

(BWR 2022, Chapter 2)

In these extensions, \(\tau\) implements time reversal by conjugation:

This is the definition of time reversal on the time side.

But we have two sides:

The mysterious 'big \(T\)' is just a representation of 'little-\(t\)', when such an extension exists.

Proposition 3.4. If a non-trivial positive-energy representation of \(\tau:t\mapsto-t\) exists on Hilbert space, then it is antiunitary \(U_\tau = T\).

Proposition 3.1. If a non-trivial positive-energy representation of \(\tau:t\mapsto-t\) exists on a symplectic manifold, then it is antisymplectic \(U_\tau = T\).

(BWR 2022, Chapter 3)

The problem of defining time reversal on a state space is just the problem of extending a representation of time translations \(G\) to a representation of \(\tilde{G}\) including time reversal \(\tau\in \tilde{G}\).

When this is possible, symmetry of time itself is encoded in state space.

Otherwise, symmetry of time itself is violated.

Plan

1. Time reversal
2. A philosophy of time
3. An arrow of time itself
4. Misdirected arrows

"The laws of physics resemble a canon by Bach. ... They do not distinguish between left and right, nor between forward and backward movements. For a long time everyone thought it had to be like that. ... [Cronin and Fitch's] discovery... implied consequences for time reflection. At least one theme is played more slowly backwards than forwards by Nature."

—Gösta Ekspong, Nobel Prize committee

Two sides:

The mysterious 'big \(T\)' is just a representation of 'little-\(t\)', when such an extension exists.

Given a representation of time translations \(G\), a representation of time reversal may not exist. This occurs precisely when there is time-reversal symmetry violation.

• BWR (2015a, 2015b), Ashtekar (2015), BWR (2022)

"It came as a great shock that microscopic T-invariance is violated in nature, that 'nature makes a difference between past and future' even on the most fundamental level."

— Bigi and Sanda (2009) CP Violation

CP violation: \(K_L \rightarrow \pi^+\pi^-\)

No 'big \(T\)' exists on state space,
so there is no temporal asymmetry in spacetime.

Wigner (1939): Spacetime structure \(\Rightarrow\) Quantum state space
Us: Quantum state space \(\Rightarrow\) Spacetime structure

Price's Table (Price 1996, Ch.1)

The representation view: asymmetries of time are projected onto state space

Plan

1. Time reversal
2. The representation view
3. An arrow of time itself
4. Misdirected arrows

Radiation

Radiation

Planck (1897), Boltzman (1897), Ritz (1908), Einstein (1909), Ritz and Einstein (1909)

Radiation

Boundary condition misfire: Explain the time asymmetry using special initial conditions (Einstein 1909).

Radiation

Heuristic misfire: 'Explanation' of the phenomenon is only possible when it is time asymmetric.

"[T]here would be advanced waves, despite the improbability, if conditions at the center were as they are when we look at the 'normal' case in reverse: in other words, if wave crests were converging, stones being expelled, and so on. The normal case shows that the statistical argument does not exclude these things as we look toward (what we call) the past. To take it to exclude them as we look toward (what we call) the future is thus to apply a double standard, and to fall into a logical circle — to assume what we are trying to prove."

— Huw Price (1996, p.57)

Radiation

'Ritz law' Postulate a new time-asymmetric law to explain the asymmetry (Ritz 1908).

Radiation

'Ritz law' Postulate a new time-asymmetric law to explain the asymmetry (Ritz 1908).

Earman's conjecture: "[A]ny EM asymmetry that is clean and pervasive enough to merit promotion to an arrow of time is enslaved to either the cosmological arrow or the same source that grounds the thermodynamic arrow (or a combination of both). But much more work would be needed before I would be willing to make this conjecture with any confidence." (Earman 2011, p.524)"

Thermo-statistical arrow

Thermo-statistical arrow

"positive time is the direction toward higher entropy"

Reichenbach (1956, p.54)

Thermo-statistical arrow

"entropy is a concept that may be bandied about
in a totally cavalier fashion!"

Penrose (1979, p.588)

Thermo-statistical arrow

Boundary condition misfire: The statistical approach to equilibrium depends on special initial conditions like the Past Hypothesis (Callender 1999, Albert 2000).

Thermo-statistical arrow

Heuristic misfire: The observables required to define equilibrium via a course-graining require contingent extra-theoretic judgements.

Thermo-statistical arrow

Missing information misfire: Zwanzig (1960) style master equations produce a time asymmetry by projecting out information.

Thermo-statistical arrow

Missing information misfire: Zwanzig (1960) style master equations produce a time asymmetry by projecting out information.

Thermo-statistical arrow

Heuristic misfire: Orthodox thermodynamic asymmetry arises only via the ad hoc addition of a new 'Minus First Law' (Brown and Uffink 2001).

Thermo-statistical arrow

Heuristic misfire: Orthodox thermodynamic asymmetry has does not exist in the structure of thermodynamic states (BWR Chapter 6).

GRW/CSL: This would provide an arrow, insofar as it is a representation of time translations that does not extend to one that includes time reversal.