The Reflective Path Integral

Mirror, Mirror
Mirror, Mirror (Star Trek: TOS)

We’ve seen signs of a mirror-image universe that is touching our own. New experiments are revealing hints of a world and a reality that are complete reflections of ours. … There might even be a race of mirror humans trying to work out why their dark matter is five times less abundant than their normal matter.
New Scientist via Futurism

The theory of mirror matter predicts a hidden sector made up of a copy of the Standard Model particles and interactions but with opposite parity.

In the original Star Trek episode “Mirror, Mirror,” the crew of the Enterprise are accidentally transported to a parallel universe. Dubbed the Mirror Universe, its denizens are evil doppelgangers of the crew, complete with garish uniforms, Nazi-like salutes, and full, robust goatees.

Like many concepts first imagined in science fiction, the mirror universe may actually exist, albeit in a far less melodramatic form.

One reason the mirror universe idea is so appealing is the math. Some models suggest a mirror universe would have to have been much cooler than our own during its early evolution. This difference would have made it easier for particles to cross over, resulting in five mirror particles for every regular one. That’s roughly the ratio of dark to normal matter.
Big Think



Do we live in a mirrorverse?

Abstract: A proto-formulation of QM (Reflective Path Integral, Sums over Histories and Futures) is proposed combining:

  • a hypothetical CPT-reversed world (“orange”) that “mirrors” our world (“blue”)
  • a reflective Feynman path ontology
  • some language syntax and semantics from Stochastic Prolog and Concurrent Prolog resulting in a Stochastic Concurrent Prolog – called σCP – for expressing QM experiments as σCP programs
  • stochastic concurrent processes path, path_integral, σ_unify
  • hidden variables as (logic programming) logical variables
  • the orange world influences the blue world stochastically via hidden variables

The EPR and Double-Slit experiments are presented in this formulation.

See also The Wheeler-Feynman-von·Neumann computer.


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 below under reconstruction.

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: [1][2].


numbers, literals ‘Y’, ‘N’, …
variables: italics x, X, …
logical variables: _X, _Y,…
particles: small letters a, b, …, x, …
“spacetime” locations: cap letters A, B, …, X, …
sets (or configurations) of locations: cap greek letters Δ, Σ, Ξ, …
path of particle x: path(x, hidden variables/state and/or locations)
path integral of x going from S via Ξ to some _D ∈ Δ:

path_integral(x, S, Ξ, Δ. _D) unifies _D with a randomly selected element of Δ 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), unifies _D with a random element of Ξ.

(In this version, Ξ is a simple set of locations. In a future version, the idea is to extend Ξ to be a configuration of locations, like a HyperDiamond Feynman Checkerboard in 4-dimensional Spacetime.)


stochastic unification of _X with respect to a distribution D:

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 process in which _X? appears is suspended until _X is bound. (See below.)

σCP, like other CPs (Concurrent Prologs), is a process-oriented programming language.

In the mirrorverse model, the concept of hidden variables being logical variables is critical. I don’t think logical variables have been used in the language of physics before.


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).

The orange world is the mirror world of the blue world, and vice versa.
Together they make the mirrorverse.

(x can be either a boson or a fermion, cf. Time reversal violation is not the ‘arrow of time’: “We could simply choose to define what we mean by “time reversal” as what most textbooks now define as ‘CPT’. Then time reversal would be a good symmetry of nature! You can actually prove that any theory that is fundamentally reversible (unitary, information-conserving) will have an operation corresponding to time reversal that is a good symmetry. So the carefully posed physics question is not “why is T violated?”, but “why is the preserved notion of time reversal one that involves what we label C and P as well?” I.e., there may be T-asymmetry in our world and the opposite T-asymmetry in the CPT-reversed world, but taking the worlds together as a whole, you get T-symmetry back. Also see Victor J. Stenger’s biverse, and this proposal by Bob Zannelli.)

For a particle x beginning at A and terminating at B, blue paths are summed at B, orange paths are summed at -A. The blue x and the orange -x never meet though: x and -x go along their respective paths in separate wolds. The hidden variables though are unified (in the logic programming sense): If _Xblue is a hidden variable in the blue world and _Xorange is the corresponding hidden variable in the orange world, then _X = _Xblue = _Xorange. Also, as I hope to make clear, all unifications occur locally (operationally at the location they occur). Thus RPI is a local theory.

(The retrocausality – or better: retrodependency – 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 particles a and b that travel from emitter S to detectors A (in one direction) and B (in the opposite direction) respectively. (See picture.) 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.)

a and -a (and b and -b) never “meet”. They just share hidden (logical) variables.

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:


From here on, ‘state’ will be used for ‘hidden variables’ as they become unified with concrete (ground term) values.

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.


_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.


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.


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

σ_unify(_Qa, {1/.333..,2/.333..,3/.333..})
σ_unify(_Qb, {1/.333..,2/.333..,3/.333..})
σ_unify((_S1,_S2,_S3,_Qa?,Qb?), D)

Note: This example is updated in σCP – Stochastic Concurrent Prolog.

So where is the “Feynman sum” of paths in this example? The paths of a and b from the source are “reflected” in the paths of -a and -b which arrive at the source: The “sum” occurs at the source!


The Double-Slit 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,_Slit?,_S?) goes from E to Σ . It is not determined what S in Σ x goes to or what Slit(r) x goes through.


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


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 the Double-Slit experiment can be written as 4 σCP processes:


Thus only one path is taken in each world. In this sum-over-histories-and-futures interpretation, “decoherence” is defined as all histories/futures but one die. (In a Darwinian”survival-of-the-fittest analogy, selection is made from a “fitness” probability distribution.)

“It is best not to think of the paths as places where there are particles. The paths are parametrized representations of quantum amplitudes, which do correspond to a particle probability[.]
– Lawrence Crowell

Retrocausality comes as a “byproduct” of the (mirrorverse) Reflective Path Integral σCP model. The Path Integral goes by different alternative names: sum over x, where x is paths, histories, possibilities, or … propensities!.

blue world with T→
orange world with ←T


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 stochastically influences (via hidden variables) our world, but we cannot talk to it.

I just posit a mirrorverse (with Reflective Feynman Paths) and logical variables (from Stochastic Concurrent Prolog – σCP) as hidden variables and “retrocausality” is the result. (You don’t have to mention that word.)

What underlies (substrates) reality is a quantum reality composed of “paths” in a Reflective Path Integral in a survival-of-the-fittest type of struggle for existence.


Perhaps the chapters of the book of nature, traditionally written in a mathematical language with its denotational-extensional semantics, are better written in a programmatical language, with its operational-intensional semantics.


Note: This post contains the preliminary definition of σCP. σCP – Stochastic Concurrent Prolog has the most most-recent version and updates to the examples of this post.

It is clear that hidden-variable interpretations of QM must have “variables” that effectively transmit information from the future to the present. The Reflective Path Integral is one of those. Another is

The Cellular Automaton Interpretation of Quantum Mechanics
Gerard ‘t Hooft


The Big Bang … also generated a second “anti-universe” that extends backwards in time, like a mirror image of our own.

[If one views QM as a generalized measure on a space of histories, then one sees not only how quantal processes differ from classical stochastic processes (the main difference, they satisfy different sum rules), but also how closely the two resemble each other.]
via Rafael Sorkin


Philip Thrift


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