(Not to be confused with Mirror, Mirror – Star Trek: The Original Series, first airdate October 6, 1967.)

In what follows, the goal is to define a *Reflective Path Ontology* leading to a **Reflective Path Integral** formulation.

(“Reflective” is defined below. Alternatives: sum over reflective histories, sum over histories and futures.)

From **Physics of the Impossible**: *A Scientific Exploration Into the World of Phasers, Force Fields, Teleportation, and Time Travel*, by Michio Kaku:

T-reversal by itself violates the laws of quantum mechanics, but the full CPT-reersed universe is allowed. This means that a universe in which left and right are reversed, matter turns into antimatter, and time runs backward is a fully acceptable universe obeying the laws of physics! (Ironically, we cannot communicate with such a CPT-reversed world. If time runs backward on their planet, it means that everything we tell them by radio will be part of their future, so they would forget everything we told them as soon as we spoke to them. So even though the CPT-reversed universe is allowed under the laws of physics, we cannot talk to any CPT-reversed alien by radio.)

*What follows is a very sketchy draft, unlikely to be believed. Hopefully, better versions to follow.*

This is the main ontological thesis: For a retrocausal reality, there would be a mirror (parallel) universe that is CPT-reversed from our universe – a universe which (stochastically) influences but does not appear in ours.

By *retrocausal*, I mean it as described in Taming the quantum spooks (Aeon). By *CPT-reversal* – or *CPT-reflection* – I mean the *simultaneous* reversal of charge conjugation (C), parity (P), and time (T). cf. clock in a mirror.

Notation:

numbers, literals (‘Y’,’N’), …

variables: italics *x*, *X*, …

logical variables: _ (‘anonymous’), _X, _Y, …

particles: small letters a,b, …, x, …

locations: cap letters A,B, …, S, …

sets of locations: cap greek letters Δ, Σ, Ξ, …

path of particle x: path(x, *state* and/or *locations*)

path integral of x going from S to some D ∈ Δ:

path_integral(x, S, Ξ, Δ) returns a D ∈ Δ randomly selected according to the Feynman probability calculus of a particle x going from S to some location in Δ via some location in Ξ. If Δ is a singleton (set with one element), returns a random element of Ξ.

stochastic unification of _X with respect to a distribution:

σ_unify(_X,

*D*) results in a single unification chosen stochastically from a population of possibilities:

A logical variable _X is unified with a single *x*[*i*] with probability *p*[*i*] from a distribution (a set with probabilities assigned to its elements)

*D* = { *x*[1]/*p*[1], …, *x*[*n*]/*p*[*n*] }.

(cf. Prolog syntax and semantics and Unification in logic programming. Stochastic unification (see below) is used in Stochastic Prolog.)

In addition: Prolog-style terms. unify is extended to terms. E.g.

σ_unify( (_X,_Y), {(‘Y’,’Y’)/.25,(‘Y’,’N’)/.25,(‘N’,’Y’)/.25,(‘N’,’N’)/.25}’

results in (_X,_Y) bound to one of the elements with equal probability.

Also, the question mark ‘?’ marks a logical variable _X: _X? (imported from Concurrent Prolog). A function or process in which _X? appears is suspended until _X is bound. (See below.)

A particle x goes from A to B. Color all Feynman Paths (FPs) going from A to B blue. Their CPT-reversed versions of paths of -x going from -B to -A are colored orange.

Entities *x* in our world, to be called the blue world, have counterparts –*x* in the orange – or mirror – world. -(-*x*) = *x*. (Blue and orange happen to be opposites on the RYB color wheel.)

Define RFPs (Reflective Feynman Paths):

RFPs = the set of all Feynman Paths + the set of all CPT-reversed Feynman Paths

The first part of the sum is in Ω = our world, the second is in -Ω = the CPT-reversed world. Call Ω = blue world, -Ω = orange world. In -Ω there would be -A and -B, the orange duplicates of A and B.

So corresponding to each RFP blue path path(x) of a particle x there is its CPT-reversed RFP orange path path(-x).

(x can be either a boson or a fermion.

cf. preposterousuniverse.com/blog/2012/11/20/time-reversal-violation-is-not-the-arrow-of-time.)

For a particle x beginning at A and terminating at B, blue paths are summed at B, orange paths are summed at -A. (Is there a possibility that the orange summation at A can stochastically influence which decision x makes at A on which path to pursue? And vice versa.) The blue x and the orange -x never meet though: x and -x go along their respective paths in separate wolds. (The retrocausality in this scenario means that although there is a “sum over histories” and “sum over futures”, only one path is selected: the one path that was already chosen probabilistically in the past based on its future. See below.)

**The EPR experiment**

There is a source S that simultaneously emits two entangled particles a and b that travel from emitter S to detectors A (in one direction) and B (in the opposite direction) respectively. In the orange world, there are the counterparts to S, A, B: -S, -A, -B.) There are not two, but four RFPs to consider: path(a), path(-a), path(b), path(-b). path(a) and path(b) are in our time perspective, path(-a) and path(-b) in the CPT-reversed perspective: Orange particles going from -A and -B arrive at -S at the same time. (In the orange world, -a and -b are absorbed by -S.)

*Note*: “Bell’s Theorem requires the assumption that hidden variables are independent of future measurement settings.” – Backward causation, hidden variables and the meaning of completeness, Huw Price. But this assumption is ruled out here, so particles will have hidden variables.

The example used here is from Huw Price’s **Time’s Arrow and Archimedes’ Point**: *New Directions for the Physics of Time* (beginning pg. 213) about what happens on a planet called Ypiaria (“Pronounced, of course, ‘E-P-aria’.”)

The scenario here is that there is a pair of twins a and b who depart from S and travel to A and B respectively. At each place A and B, there is an interrogator who asks them respectively a question.

One question only could be asked, to be chosen at random from a list of three:

(1) Are you a murderer?

(2) Are you a thief?

(3) Have you committed adultery?

The assumption is that each twin is truthful. The interrogators recorded all questionings of all twin pairs.

The records came to be analyzed by the psychologist Alexander Graham Doppelganger.

He found that

(D-1) When each member of a pair of twins was asked the same question, both always gave the same answer; and that

(D-2) When each member of a pair of twins was asked a different question, they gave the same answer on close to 25 percent of such occasions.

It may not be immediately apparent that these results are in any way incompatible.

What follows in Price’s Ypiaria story is how Doppelganger reasoned this out. (This is related to statistics to a real EPR experiment.) Below is how it could work out in a Reflective Path Integral (RPI) formulation.

Let *S*(1) = ‘Y’ or ‘N’, *S*(2) = ‘Y’ or ‘N’, *S*(3) = ‘Y’ or ‘N’ (corresponding to “Yes” or “No” responses).

a with hidden variables (_S1,_S2,_S3):_Qa is sent from S to A; b with hidden variables (_S1,_S2,_S3):_Qb is sent from S to B. This is represented as two paths:

path(a,(_S1,_S2,_S3):_Qa)

path(b,(_S1,_S2,_S3):_Qb)

From here on, ‘state’ will be used instead of ‘hidden variables’.

At -A: σ_unify(_Qa, {1/.333..,2/.333..,3/.333..})

At -B: σ_unify(_Qb, {1/.333..,2/.333..,3/.333..})

-a with state (_S1,_S2,_S3):_Qa = antiparticle returned from -A in reverse time, -b with state (_S1,_S2,_S3):_Qb = antiparticle returned from -B in reverse time.

path(-a,(_S1,_S2,_S3):_Qa)

path(-b,(_S1,_S2,_S3):_Qb)

_Qa, _Qb each are bound to 1,2,or 3. Let *Aq* = _Qa, *Bq* = _Qb. (_Qa and _Qb are ground terms.)

In the the orange world, -a and -b are absorbed by -S. This stochastically influences the distribution of (*S*(1),*S*(2),*S*(3)):_ in the blue world, as follows.

Assign probabilities to each possibility:

prob( (*S*(1),*S*(2),*S*(3),*Aq*,*Bq*) )

There are 2*2*2*3*3 = 72 possibilities. Let *P* be this set.

Let

*Q* = { (*S*(1),*S*(2),*S*(3),*Aq*,*Bq*) | *Aq* != *Bq*, *S*(*Aq*) = *S*(*Bq*) }

It turns out that *Q* represents 24 of the 72 possibilities in *P*.

prob( (*S*(1),*S*(2),*S*(3),qA,qB) ) is defined such that

prob(*Q*) = .25 (should be .333… if all possibilities are equally likely)

It turns out that the probability to be assigned to each of the members of the *Q* sum is about .01, and the all the other possibilities about .0158.

Define a distribution *D* over *P* with elements assigned these probabilities.

Then σ_unify( (_S1,_S2,_S3, *qA*, *qB*), *D*).

This selects *S*(1),*S*(2),*S*(3) in the blue world.

Hidden variables shared between blue and orange worlds.

Introduce ‘?’ (Concurrent Prolog) for “suspend until binding”.

path becomes a process.

The EPR experiment can then be written as 7 concurrent processes:

path(a,(_S1,_S2,_S3):_Qa)

path(b,(_S1,_S2,_S3):_Qb)

σ_unify(_Qa, {1/.333..,2/.333..,3/.333..})

σ_unify(_Qb, {1/.333..,2/.333..,3/.333..})

path(-a,(_S1,_S2,_S3):_Qa?)

path(-b,(_S1,_S2,_S3):_Qb?)

σ_unify((_S1,_S2,_S3,_Qa?,Qb?), *D*)

**The double-split experiment**

There are four paths to consider for particle x that goes from the emitter E to screen Σ (to a screen location *S* ∈ Σ where there is a panel with two slits Slit(1) and Slit(2) between the emitter and screen:

path(x,E,Slit(1), *S*)

path(x,E,Slit(2), *S*)

path(-x,-*S*,-Slit(1), -E)

path(-x,-*S*, -Slit(1), -E)

1. path(x,E,_,_) goes from E to Σ . It is not determined what *S* in Σ x goes to or what Slit(*r*) x goes through.

*S* = path_integral(x,E,{Slit(1),Slit(2)},Σ)

2. path(-x,-*S*,_,-E) goes from -S to -E in the orange world. The slit is not determined.

path(-x,-*S*,-Slit(1),-E) and path(-x,-S,-Slit(1),-E) are the possible orange paths from –*S* to -E.

Let -Slit(*r*) = path_integral(-x,-*S*,{-Slit(1),-Slit(2)},{-E}), *r* = rand(1,2).

Then in the blue world, x takes the path from E to Σ that goes to *S* through Slit(*r*).

3. path(x,E,Slit(*r*),*S*) is taken in blue world.

4. path(-x,-*S*,-Slit(*r*),-E) is taken in orange world.

So

*S* = path_integral(x,E,{Slit(1),Slit(2)},Σ)

path(x,E,-path_integral(-x,-*S*,{-Slit(1),-Slit(2)},{-E}),*S*)

is the path taken by x.

(Only one path is taken in each world.)

_X = *x* is equivalent to σ_unify(_X,{*x*/1.0}).

_S = path_integral(x,E,{Slit(1),Slit(2)},Σ)

path(x,E,-path_integral(-x,-_S?,{-Slit(1),-Slit(2)},{-E}),_S?)

*Need to go back and clean up the two experiment sections.*

Review:

Two stochastic operators were introduced: **path_integral** and **σ_unify**. Both operate in forward and backwards worlds.

Can *reflective path integration* with *stochastic logical unification* be the ingredients for a retrocausal reality?

The full picture of the cosmos is this: There is the whole collection of RFPs, the constituents of an RPI reality – our perspective and a CPT-reversed perspective – but we can only (fully) experience “half” of it. The CPT-reversed world is an orange ghost that influences our world, but we cannot talk to it.

Philip Thrift