The Transactional Interpretation of Quantum Mechanics

Today I have the privilege to host a guest blog post by Ruth E. Kastner (@rekastner http://transactionalinterpretation.org/ ) who is an expert in the Transactional Interpretation of Quantum Mechanics. She recently published a book on this topic.

Part 1: Introduction

            The Transactional Interpretation of Quantum Mechanics (‘TI’ for short) was proposed by John G. Cramer in the 1980s (Cramer’s comprehensive 1986 presentation is here). The distinguishing feature of TI is its ability to ‘read off’ the Born Rule for the probabilities of measurement results from the basic formalism. How does it do this? By including the solutions of the advanced (negative energy/time reversed) Schrödinger Equation, as well as the retarded solutions of the usual Schrödinger Equation.

            TI is based on the Wheeler-Feynman time-symmetric theory of classical electromagnetism (Wheeler and Feynman 1945, 1949). In this theory, a charge emits a time-symmetric field having equal components retarded and advanced radiation, while any absorber responds to that radiation by generating its own time-symmetric field. The responding field is exactly out of phase with the stimulating field. This results in the cancellation of all fields except the one between the emitter and the absorber—that field is built up to full strength, thereby transporting energy from the emitter to the absorber. (See below.)

            The Wheeler-Feynman theory accounted nicely for ‘radiative damping,’ or loss of energy by a radiating charge. This could only be accounted for in the traditional theory (which assumes only a retarded field is radiated) by assuming an ad hoc sourceless ‘free field’.  Wheeler and Feynman’s original motivation in developing this ‘direct action’ theory was to eliminate the field as an independently existing entity, to solve the problem of self-action in the classical theory – in which a charge interacted with its own field, resulting in infinite energies. They lost interest in it when they found that some degree of self-action was needed to account for relativistic effects (such as the Lamb shift). However, the direct action picture can still be used, and in fact Davies developed a direct-action picture of quantum electrodynamics in the 1970s (Davies 1970, 1971, 1972).[1]

            The transactional interpretation is an application of the direct action picture to quantum systems. It departs from the Wheeler-Feynman (WF) classical picture in the fact that, instead of being a continuous quantity, energy can only be delivered in discrete packets (quanta). The usual retarded quantum states, such as |Y>,  are called offer waves (OW) and the advanced states, <F|,  are called confirmation waves (CW). Since TI deals with quantized energy and momentum, many absorbers may respond with CWs to an emitter’s OW, but (for a one-quantum OW) only one of the responding absorbers can actually receive the energy. But this quantized version of the WF picture allows for a very nice resolution of the notorious ‘problem of measurement’: namely, it allows one to ‘read off’ the heretofore mysterious ‘Process 1’ of John Von Neumann’s theory of measurement, which also contains the Born Rule. We’ll now see how this works.

            To review, ‘Process 1’ is the transition of a quantum system from a pure state to a mixed state upon measurement, i.e.:

The coefficients |c_n|² are the probabilities given by the Born Rule for each of the outcomes y_n.

Von Neumann noted that this transformation is acausal and fundamentally irreversible, yet he was unable to explain it in physical terms, and treated this transition as fundamentally dependent on an observing consciousness. However, if we take into account the advanced responses of absorbers, then for an OW described by |Y>, we have for a collection of numbered absorbers:

In the above diagram, an initial offer wave from emitter E passes through some measuring apparatus that separates it into components áy_n|Yñ |y _nñ, each reaching a different absorber n. Each absorber responds with an advanced (adjoint) confirmation áy_n| áY|y _nñ. In TI, these OW/CW encounters are called incipient transactions. They are described in probabilistic terms by the product of the OW and CW, which gives a weighted projection operator:  áy_n|YñáY|y_nñ |y _nñáy _n | = |c_n|² |y _nñáy _n |. If we add all the incipient transactions, we clearly have the density operator representation of ‘Process 1”:

Thus, by including the advanced responses of absorbers, we have a physical account of measurement as well as a natural explanation of the Born Rule and Von Neumann’s ‘Process 1’. The response of absorbers is what creates the irreversible act of measurement and breaks the linearity of the basic deterministic propagation of the quantum state. Since the conserved physical quantities can only be delivered to one absorber, there is an indeterministic collapse into one of the outcomes y _k  with a probability given by the weight |c_k|²  of the associated projection operator |y _kñáy _k |. This is called an actualized transaction, and it consists in the delivery of energy, etc., to one absorber. I see this is a form of spontaneous symmetry breaking (this is discussed in Chapter 4 of my book).

            I have been developing the transactional picture in two specific ways: (1) investigating what kind of ontology underlies the offers and confirmations, especially when these are multi-quantum states: and (2) extending TI explicitly into the relativistic domain to give a more detailed account of emission and absorption, both of which are inherently relativistic processes. Both emission and absorption play crucial roles in defining the measurement process.  Only the transactional picture, in which absorption contributes advanced states, provides a physical basis for what is otherwise an ad hoc mathematical recipe (Von Neumann’s ‘process 1’ and the accompanying Born Rule). In Part 2, I will discuss my recent developments of TI. Further information and discussion is available at my website (http://transactionalinterpretation.org/ ).

References

Cramer J. G. (1986). ``The Transactional Interpretation of Quantum Mechanics.''  Reviews of Modern Physics 58, 647-688.

Davies, P. C. W. (1970). “A quantum theory of Wheeler-Feynman Electrodynamics,” Proc. Cam. Phil. Soc. 68, 751.

 ___________(1971).”Extension of Wheeler-Feynman Quantum Theory to the Relativistic Domain I. Scattering Processes,”  J. Phys. A: Gen. Phys. 6, 836.

____________(1972).”Extension of Wheeler-Feynman Quantum Theory to the Relativistic Domain II. Emission Processes,”  J. Phys. A: Gen. Phys. 5, 1025-1036.

Kastner, R. E (2012). The Transactional Interpretation of Quantum Mechanics: The Reality of Possibility. Cambridge: Cambridge University Press.Wheeler, J.A.  and R. P. Feynman, "Interaction with the Absorber as the Mechanism of Radiation," Reviews of Modern Physics, 17, 157–161 (1945).

Wheeler, J.A.  and R. P. Feynman, "Classical Electrodynamics in Terms of Direct Interparticle Action," Reviews of Modern Physics, 21, 425–433 (1949).

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[1] Meanwhile, the infinities resulting from self-action are being ‘tamed’ by renormalization techniques.

Quantum Teleportation

One of the important results in quantum mechanics is the teleportation protocol. There are several misconceptions about teleportation, fueled in part by Star Trek episodes.

For example, you cannot teleport faster than the speed of light. Quantum mechanics exhibits correlations above what can be expected by a causal local theory and an experiment here can obtain data instantaneously correlated with an experiment located at the other side of Milky Way. To verify the correlation however, one needs to physically travel from here to there and this can only be done at speeds below the speed of light. Quantum mechanics is non-local, but it does not permit violation of special theory of relativity. Just because one measures something over here it does not carry any instantaneous signals across the galaxy (this is usually called no-signaling).

Then, unlike in Star Trek, the teleported object does nor vanish. Instead, all its information is extracted and in the process the object gets destroyed. From the extracted information, the quantum state is reconstructed at the destination using local materials. In a way, it is like breaking up an apply pie to extract the baking instructions, and then using those instructions re-baking the pie at the destination using local ingredients. If you are teleported, you will die on the teleporting pad, and hopefully your complete atom state information is transmitted and used to make up an exact replica of yourself at the destination. A few mistakes in measuring and reconstruction and you will end up like someone from a Picasso painting-not an appealing prospect.

So what is the big deal then with quantum teleportation? Is there any difference between a fax machine and a quantum teleportation device? In the quantum world there are two basic barriers. First, in quantum mechanics a state cannot be copied (cloned). The no cloning theorem means that a classical fax machine is impossible to be implemented in the quantum realm. The second roadblock is that measuring the quantum state destroys (or collapses) what it is measured and you cannot access all of the quantum state information. So it was a very exciting time when the teleportation protocol was discovered. This protocol can enable future quantum computers to transmit quantum information. But how does it work? We can start with the mathematical description (see here ) but it is much more entertaining to repeat a funny story originally told by Charles Bennett, one of the discoverers of teleportation. I could not find a good reference for this story, so I will tell it from memory attempting to be as close as possible with the original story (I originally blogged about this at FQXi and I had a link there to Bennett’s story but the link is now broken).

Suppose there are two brothers Romulus and Remus who don’t know much about anything. When asked about any question they answer randomly, but they both give the same answer:

Teacher: What color is the grass?

Romulus: Pink ma’am.

Another Teacher in another room: What color is the grass?

Remus: Pink sir.

Now as the story goes, a murder was committed in Boston and the FBI wants to talk with the sole witness. They do not trust the local cops, and since the witness is still in a state of shock, they cannot transport him to Washington DC to be interviewed by FBI experts because they risk tampering the brittle witness’ state of mind. Fortunately Romulus happens to be in Washington DC and Remus in Boston. Then they ask Remus to spend time with the witness talking about any topics they want: the weather last weekend, the best movie showing right now, the stock market, etc. So Remus spends an hour with the witness and at the end of the hour, the witness said he hates Remus because he dislikes every single thing Remus likes. Moreover, the stress of the meeting has completely erased his recollection of the crime. Can the FBI agents in Washington DC have any chance to find out about the crime? Surprisingly the answer is yes. They will ask Romulus about the crime (he did not witness anything; he was in Washington DC all the time) and armed with the information about the outcome of the meeting between Remus and the witness, they reverse every single the answer Romulus provides them when asked about the crime and solve the case.

Nice story? Unbelievable?

Let’s say it again using quantum mechanics mathematics this time.

Let’s meet the key people:

| ψ > = α |0> + β |1>  ------ the witness state to be teleported from Boston to DC

| Φ+> = 1/sqrt(2) [|0>⊗ |0> + |1>⊗ |1>] – maximal entangled state -------Romulus-Remus pair

The total state is:

| Φ+>⊗|ψ> = 1/sqrt(2) (0>⊗ |0> + |1>⊗ |1>)⊗ (α |0> + β |1>)

Let’s introduce 4 completely entangled states:

| Φ+> = 1/sqrt(2) [|0>⊗ |0> + |1>⊗ |1>]

| Φ-> =  1/sqrt(2) [|0>⊗ |0> - |1>⊗ |1>]

| Ψ+> = 1/sqrt(2) [|0>⊗ |1> + |1>⊗ |0>]

| Ψ-> =  1/sqrt(2) [|0>⊗ |1> - |1>⊗ |0>]

Then:

|0>⊗ |0> = 1/sqrt(2) [|Φ+> + | Φ->]

|0>⊗ |1> = 1/sqrt(2) [|Ψ+> + | Ψ->]

|1>⊗ |0> = 1/sqrt(2) [|Ψ+> -  | Ψ->]

|1>⊗ |1> = 1/sqrt(2) [|Φ+> - | Φ ->]

and therefore

| Φ+>⊗|ψ> = 1/sqrt(2) (0>⊗ |0> + |1>⊗ |1>)⊗ (α |0> + β |1>) =

½( α |0>r⊗ |0>r⊗|0>w  + β |0>r⊗ |0>r⊗|1>w  + α |1>r⊗ |1>r⊗|0>w  + β |1>r⊗ |1>r⊗|1>w)

where the indices r,r,w represent Romulus, Remus, and the witness. Now we will swap Romulus to the right with witness to the left and after a bit of elementary algebra we get the expression above written in this form:

½ (|Φ+>rw⊗(α |0>r + β |1>r) + |Φ->rw⊗(α |0>r - β |1>r) + |Ψ+>rw⊗(β |0>r + α |1>r) + |Ψ->rw⊗(β |0>r - α |1>r))

This rewrite is the clever idea of teleportation.

The Boston meeting between Remus and the witness corresponds to a measurement for the Remus-witness part which will collapse the state to one of the 4 possible outcomes:

|Φ+>rw

|Φ->rw

|Ψ+>rw

|Ψ->rw

Consequently the state of Romulus in Washington DC is one of the four possibilities:

α |0>r + β |1>r

α |0>r - β |1>r

β |0>r + α |1>r

β |0>r - α |1>r

Then the measurement outcome is sent by classical means to Washington DC. (OK, the story is a bit more complex than the Romulus-Remus story above, in the real protocol there are 4 outcomes)

All that is left to do is to apply a local transformation (like reversing the answers from Romulus) in Washington DC to transform Romulus state to:

α |0> + β |1>

thus achieving the teleportation of the unknown original state from one place to the other. In the process we did not copy the state (the original state got destroyed) and the no-clone theorem was obeyed. Also we did not extract the coefficients α and β like in a classical faxing process. The memory of crime got teleported from the witness into the head of Romulus who for the purposes of the FBI investigation has become the witness. This was achieved by what is called quantum steering: measuring on one entangled particle changes remotely the state of its pair. This is instantaneous and can be done from one end of the galaxy to the other if you like, but in the absence of the right key to unlock the information it is useless. The key still has to travel slower than the speed of light and relativity is ultimately obeyed.  

The Kochen-Specker Theorem

I will now take a break from the prior series, and discuss a few fundamental results in quantum mechanics which were not given proper attention in the past on this blog. Today I am discussing the Kochen-Specker theorem, which rivals in importance Bell’s theorem.

When you read about this result the first time, it looks a bit dry and abstract, but in fact it is child’s play because it is nothing more than a coloring game. Originally the first proof was quite intricate, but later on, the late Asher Peres found a great simplification and I will discus this instead.

Before starting, we need one preliminary result: for particles of total spin 1, we can measure the square of the component of spin in a direction and get +1 or 0. So far nothing special, but quantum mechanics shows that if we perform such a measurement on three orthogonal directions (say on x, y, z) we will get two results of +1 and one result of 0. We do not know what result will be on which direction, but we will always get a zero result and two +1 results in some order. I’ll not prove this, but I want to look at its meaning instead.

We know that in quantum mechanics the results of experiments do not exist before measurement, but can we create a model which will recover the 1,1,0 prediction for spin one particles? This will work for three orthogonal directions, but what if are adding additional orthogonal directions? If particles do have definite properties before measurement, it should be possible in principle to pre-assign the values of +1 and 0 to all our measurement directions in such a way that the +1,+1, 0 theorem is obeyed. The Kochen-Specker theorem shows that this is impossible.

Let us start with the measurement directions that Peres found:

(I am adapting this from a famous paper about Free Will: http://arxiv.org/pdf/quant-ph/0604079v1.pdf )

So here is the explanation: start with a cube, inscribe a circle on each face and add a point on the 4 places it touches the face sides (e.g. points V,D, U, etc). Then add a point in the middle of each face (points X, Y, Z) and connect this with the vertexes of the square. Add a point where this line intersects the circle (C, B, C’, B’) and unite them in a smaller square. Draw perpendiculars from the center of the face to the small square and add 4 more points (e.g. D, D’). Finally connect the center of the cube with all those points and obtain 33 directions: 13*3 – 4*3/2 = 33 directions [13 point on a face*3 faces but all 4 points on the inscribed circle are counted twice]

Now we can start the coloring game and prove the Kochen-Specker theorem. We will color the 1’s as red, and the 0’s as blue and see if this can be done in general or not.

Step 1: X, Y, Z for 3 orthogonal directions:

(we can pick X as zero without loss of generality)

Step 2: X-A implies A = +1 because the center of the cube and X and A form orthogonal directions, and on any 3 orthogonal directions there can be only one zero which is at X now) In turn A’ = 1 because X, A, A’ form 3 orthogonal directions.

Step 3: A,B,C form 3 orthogonal directions (this shows the cleverness of Peres’ choice for directions, try to prove using simple geometry that A,B,C form 3 orthogonal directions). Without loss of generality we can pick B=1, C=0.

and from A’ B’ C’: B’ = 1 and A’ = 1

Step 4: orthogonality of CD implies D = 1. Similarly C’D’ implies D’ = 1

Step 5: Z, D, E orthogonality implies E = 0. Z, D’, E’ implies E’ = 0

Step 6: EF and EG orthogonality implies G=G = 1. E’F’ and E’ G’ implies F’ = G’ = 1

Step 7: F, F’, U implies U = 0

Step 8: G, G’, V implies V = 0

Now for the contradiction: U is orthogonal with V and you cannot have both of them equal with zero.

So what does this mean? This shows that we cannot have a context independent assignment of measurement outcome before measurement. If we want to pre-assign measurement properties, this can only be done within the context of the measurement setting. K-S theorem weakens the idea of objective reality independent of measurement.

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Solving Hilbert’s Sixth Problem

Tying it all together for quantum mechanics

Using commutators and anti-commutators we have seen last time the relationship between the two products. The remarkable fact is that this relationship can be derived from composability (see http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.cmp/1103900192 for the lengthy proof) and in general you get:

[A, B, C]_θ + ħ ² /4[A, B, C]_xρ =0

where [A, B, C] = (AB)C – A(BC) is the associator

with x=+1, 0, -1

In the quantum mechanics case (x=-1), since ρ is skew-symmetric:

[A ρ A, B, A] _ρ = ((A ρ A) ρ B) ρ A – (A ρ A) ρ (B ρ A) = ((0) ρ B) ρ A – (0) ρ (B ρ A)

=0

and therefore:

[A θ A, B, A]_θ =((A θ A) θ B) θ A – (A θ A) θ (B θ A) = 0

which shows that θ is a Jordan product!!!

Since ρ is skew-symmetric and obeys the Leibniz identity:

A ρ (B ρ C) = (A ρ B) ρ C + B ρ (A ρ C) it is easy to show that it obeys the Jacobi identity:

A ρ (B ρ C) = (A ρ B) ρ C + B ρ (A ρ C) = -C ρ (A ρ B) - B ρ (C ρ A)

and so:

A ρ (B ρ C) + B ρ (C ρ A) + C ρ (A ρ B) = 0 [Jacobi identity]

Hence ρ is a Lie algebra.

In turn the Jordan and Lie algebra give rise to a C* algebra and we obtain quantum mechanics in the algebraic formalism. The standard Hilbert space formulation is recovered by the GNS theorem/construction.

In the classical case (x=0) there are no Jordan algebras, and in this case one has the regular function multiplication and the Poisson bracket as realizations of the products θ and ρ.

What can we say about the third case, the hyperbolic composability x=+1?

In this case we are lead to a hypothetical quantum mechanics over split complex numbers.The interesting part is that in this number system, the functional analysis is completely changed because the norm triangle inequality which is the foundation of most of the results in functional analysis is replaced by a reversed triangle inequality (http://arxiv.org/pdf/1311.6461v2.pdf ). The key difference however between complex quantum mechanics (parabolic composability) and split-complex quantum mechanics (hyperbolic composability) is the lack of positivity. In other words, we are not guaranteed to have positive probability predictions, and we cannot define probabilities!!! Hyperbolic composability violates one of the principles of nature introduced in prior posts: positivity. Mathematically hyperbolic quantum mechanics is just as rich and interesting as ordinary quantum mechanics, but it cannot correspond to anything in nature. Only parabolic composability (classical mechanics) and elliptic composability (quantum mechanics) can describe nature.

But how can we tell classical and quantum mechanics apart? Simple: by experimental evidence in the form of violations of Bell inequalities. In classical mechanics, x=0 which means that the ontology always factorizes neatly into system A and system B, but because x=-1 in quantum mechanics, this factorization is no longer possible, and this is known as entanglement due to the superposition of the wavefunction. It is superposition which allows for higher correlations than what one can expect from any local realistic model.

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Solving Hilbert’s sixth problem

Commutative vs. non-commutative geometry

Last time I introduced the fundamental relationship of (Hamiltonian) mechanics:

ρ₁₂= ρ⊗θ +    θ⊗ρ

θ₁₂ = θ⊗θ + x ρ⊗ρ

This looks very abstract, and nothing like the usual formulation but we will show that this leads to classical and quantum mechanics.

However, we want to take a closer look first at the relationship and notice a very interesting analogy. Suppose x = -1.Where have we seen a relation like this before? How about complex number multiplication?

z = a+ib, w = p+iq

z*w = ap - bq + i (aq + bz)

Im(zw) = Im(z) Re(w) + Re(z) Im(w)

Re(zw) = Re(z) Re(w)  - Im(z) Im(w)

Since x=-1 corresponds to quantum mechanics, it is no wonder that quantum mechanics is expressed best over complex numbers!

In general the products ρ and θ can be proven to be skew-symmetric and symmetric but I am going to skip the proof. What I want to concentrate today is the concrete realizations of the two products which is a very interesting story with unexpected mathematical links.

It turns out that there are two such realizations, one based on state space, and one based on Hilbert space. However, those realizations tell a much larger story, that of commutative and non-commutative geometry.

In mathematics there is a nice duality between geometry and algebra. The easiest way to see this is to consider the ordinary 2D plane and old fashion Euclidean geometry in a plane. Then add a coordinate system and express all lines and circles as algebraic relationships. For example a circle with center at position (a, b) and radius r obeys (x-a)^2+ (y-b)^2 = r^2. And then all usual geometric theorems can be expressed algebraically.

This duality cuts very deep across many mathematical concepts and structures, and in particular generalizes to non-commutative geometries where the notions of point and lines are not well defined. In non-commutative geometry at core, the very notion of distance is modified from an infimum (the shortest distance between two points is a line which minimizes the distance) to a supremum.

There is a dictionary of correspondence of mathematical structures between commutative and non-commutative realm (http://arxiv.org/pdf/math/0408416v1.pdf):

Commutative	Noncommutative
measure space	von Neumann algebra
locally compact space	C∗- algebra
vector bundle	finite projective module
complex variable	operator on a Hilbert space
real variable	sefadjoint operator infinitesimal compact operator
range of a function	spectrum of an operator
K-theory	K-theory
vector field	derivation
integral	trace
closed de Rham current	cyclic cocycle
de Rham complex	Hochschild homology
de Rham cohomology	cyclic homology
Chern character	Chern-Connes character
Chern-Weil theory	noncommutative Chern-Weil theory
elliptic operator	K-cycle
spin Riemannian manifold	spectral triple
index theorem	local index formula
group, Lie algebra	Hopf algebra, quantum group
Symmetry	action of Hopf algebra

To this list on the commutative side I will add: phase space and to the noncommutative side I will add Hilbert space. I will also discuss only quantum mechanics (meaning the elliptic composability case of x=-1)

Here are the realizations of the two products

Commutative	Noncommutative
Product ρ (skew symmetric)
A ρ B=A 2/ ħ(sin (ħ/2)∇) B Moyal bracket	A ρ B = i/ħ (AB-BA)   commutator
Product θ (symmetric)
A θ B=A (cos (ħ/2)∇) B	A θ B = 1/2(AB+BA) Jordan product

Where ∇ is the Poisson bracket (where the first partial differential acts on the left and the second partial differential acts on the right).

The particular choice of realization depends on the problem under consideration and the ease of the formalism to solve it, but both realizations give the same final answer. Let us play a bit with the noncommutative side products to derive an (trivial in this formalism) identity (but which is absolutely essential).

We will use [,] as a notation for the commutator, and {,} for the anti-commutator (Jordan product). Then consider this:

[A,[B,C]] – [[A,B],C] and {A,{B,C}} – {{A,B},C}

those are called associators because they measure the lack of associativity in the products [], {}.

[A,[B,C]] – [[A,B],C] = [A, BC-CB] – [AB-BA, C]=

ABC-ACB-BCA+CBA   - ABC+BAC+CAB-CBA = BAC+CBA – ACB-BCA

{A,{B,C}} – {{A,B},C} = {A, BC+CB} – {AB+BA, C}=

ABC+ACB+BCA+CBA   - ABC-BAC-CAB-CBA = -(BAC+CBA – ACB-BCA)

In fact the two associators are proportional, and this is the consistency relationship between dynamic and ontology!!! It is this relationship that transforms the two products into a mechanic (quantum or classical)! What this shows is that the two products can be combined into an associative product. Why is this important? Because it allows for the introduction of probabilities into the formalism and the notion of a state/phase space which is needed if we have to make experimental predictions, doing physics and not pure math. In fact, the table above should be extended as follows:

Commutative	Noncommutative
Product ρ (skew symmetric)
A ρ B=A 2/ ħ(sin (ħ/2)∇) B Moyal bracket	A ρ B = i/ħ (AB-BA)   commutator
Product θ (symmetric)
A θ B=A (cos (ħ/2)∇) B	A θ B = 1/2(AB+BA) Jordan product
Associative product
f⋆g = f θ g + i ħ/2 f ρ g the start product	AB = AB (matrix multiplication) Ordinary complex number multiplication

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Solving Hilbert’s sixth problem

The fundamental relationship between ontology and dynamic

Last time we have derived the Leibnitz identity which is the root cause for information conservation in nature (from this one can derive unitarity in quantum mechanics for example). How does Leibnitz identity hold under tensor composition?

The quantum reconstruction argument is categorical, meaning it is naturally expressed in terms of category theory but whenever possible we will stress a more physical point of view. So now let us introduce a composability category U(⊗,R, ρ,…) where ⊗ is the tensor product which combines two physical systems A and B into a larger system: A⊗B. As an example, A⊗B could be a hydrogen atom, where A is the electron, and B is the proton.

R represents the real number field and we pick R over any other mathematical field because we want to be able to compute probabilities in the usual way. ρ,… belong to our unspecified set of local operations {o}. This composability category has as the identity element the chosen field R (U⊗R = R⊗U = U) and elements of R will be understood as arbitrary constant functions.

Now we can see how ρ acts on a constant function 1. Using the Leibnitz identity:

f ρ 1 = f ρ (1 x 1) = (f ρ 1) x 1 + 1 x (f ρ 1) = 2x (f ρ 1)

and so we have in general that f ρ1 = 0 for any function f. (This is very natural, we just stated in a fancy abstract way that the derivation of a constant function is zero)

Using the tensor product and using the unit of the composability category we have:

(f ⊗1) ρ₁₂ (g ⊗1) = (f ρ g) ⊗1 = (f ρ g)

where ρ₁₂ is the bipartite product ρ_.

The bipartite (and in general the n-partite) products must be build out of the available products in {o}. If in our universe of discourse the collection {o} of the available products contains only the product ρ we have:

.   (f ⊗1) ρ₁₂ (g ⊗1) = (f ρ g) ⊗(1 ρ 1) = 0 because (1 ρ 1) = 0

Hence the ρ product is trivial if it exists by itself. There must exists at least another product θ to have an interesting domain. If ρ corresponds to the dynamic θ corresponds to ontology (observables).

Suppose that {o} contains only ρ and θ. The bipartite products ρ₁₂ and θ₁₂ must be constructed out of ρ and θ. The most general way for this is as follows:

(f₁ ⊗ f₂)ρ₁₂(g₁ ⊗ g₂) = a(f₁ρ g₁) ⊗ (f₂ρ g₂) +b(f₁ρ g₁) ⊗ (f₂θ g₂) + c(f₁θ g₁) ⊗ (f₂ρ g₂) + d(f₁θ g₁) ⊗ (f₂θ g₂)

(f₁ ⊗ f₂) θ ₁₂(g₁ ⊗ g₂) = x(f₁ρ g₁) ⊗ (f₂ρ g₂) +y(f₁ρ g₁) ⊗ (f₂θ g₂) + z(f₁θ g₁) ⊗ (f₂ρ g₂) + w(f₁θ g₁) ⊗ (f₂θ g₂)

In shorthand notation:

Rho_12 = a rho_1 rho_2 + b rho_1 theta_2 + c theta_1 rho_2 + d theta_1 theta_2

Theta_12 = x rho_1 rho_2 + y rho_1 theta_2 + z theta_1 rho_2 + w theta_1 theta_2

Now the goal is to determine the values of the parameters a,b,c,d,x,y,z,w. Since the relationship is general, we can pick f₁  =f₂ = 1 and we can use the identity: 1 rho g = 0;

Then in the first relationship only c and d terms survive. If we normalize Theta such that (1 theta 1) = 1 this demands c=1 and d=0. Similarly z=0, w = 1. Using the same trick with g₁  = g ₂ = 1 demands b=1 y=0

So we must have:

(f₁ ⊗ f₂)ρ₁₂(g₁ ⊗ g₂) = a(f₁ρ g₁) ⊗ (f₂ρ g₂) +(f₁ρ g₁) ⊗ (f₂θ g₂) + (f₁θ g₁) ⊗ (f₂ρ g₂)

(f₁ ⊗ f₂) θ ₁₂(g₁ ⊗ g₂) = x(f₁ρ g₁) ⊗ (f₂ρ g₂) + (f₁θ g₁) ⊗ (f₂θ g₂)

The a term can be eliminated by applying the Leibnitz identity on itself on the bipartite products ρ₁₂. Therefore the fundamental composability relation becomes:

ρ₁₂ = ρ⊗θ + θ⊗ρ

θ ₁₂ = θ⊗θ + xρ⊗ρ

where x could be normalized to be +1, 0, -1.

The three possible parameters -1, 0, 1 corresponds to “fixed points” in a categorical (composability) theory and they correspond to (this remains to be shown):

-quantum mechanics (elliptical composability)

-classical mechanics (parabolic composability)

-split-complex (hyperbolic) quantum mechanics (hyperbolic composability)

If Theta represents the algebra of observables and Alice and Bob form a bipartite system (EPR pair), x=0 means that the observables are separable. x != 0 means that the observables are affected by the dynamics and the system can be entangled!!!

We can normalize x to be +1,0,-1, but if we do preserve the dimensions, when it is not zero, x = +/- ħ²/4.

Invariance of the laws of nature under composability demands that x remains the same or, equivalently that the Plank constant is the same for all quantum systems!

Now for some references.

The core ideas were developed by Emile Grgin and Aage Petersen at Yeshiva University in the 70s (Aage Petersen was Bohr’s assistant and Bohr had the hunch that classical and quantum mechanics share core features beyond the correspondence principle)

http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.cmp/1103900192

The idea that composability demands the invariance of the Plank constant was developed by Sahoo, a colleague of Grgin: http://arxiv.org/abs/quant-ph/0301044

Expanding on Grgin’s original idea I wrote: http://arxiv.org/abs/1303.3935 which was uploaded on the archive only 11 days before a similar result by Anton Kapustin of Caltech: http://arxiv.org/abs/1303.6917 Kapustin’s paper had the same old Grgin paper for inspiration and is written in the category theory formulation. We worked independently and the papers are about 80% identical in content notwithstanding that they look very different. Eliminating (correctly and conclusively) the hyperbolic composability class was done in http://arxiv.org/abs/1311.6461 and his lead to a major generalization of functional analysis (I’ll cover this in subsequent posts). There are still more unpublished results.

If the reader is interested into an excellent reference for classical and quantum mechanics (which includes the Lie-Jordan algebraic formulation of quantum mechanics) I highly recommend Nicolas Landsman’s book: http://www.amazon.com/Mathematical-Classical-Mechanics-Monographs-Mathematics/dp/038798318X

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Solving Hilbert’s sixth problem

Invariances of the laws of nature

How is special theory of relativity derived? One starts with the invariance of the laws of nature to changing inertial reference frames. From this you get either the Lorentz transformation or the Galilean transformation, and use the second postulate, that of the constant value of the speed of light, to select between the two choices.

But are the laws of nature invariant only to changes in inertial reference frames? How about a trivial invariance: the laws of nature do not change during time evolution?

And how about another: the laws of nature do not change if we partition in our mind a larger system into smaller ones (the composability principle, or the invariance of the laws of Nature under tensor composition)?

Those statements are completely obvious, but their mathematical consequences are far from trivial. We need one more ingredient of a technical nature: a continuity property: if we represent the state of a physical system with a point in a state/phase/configuration space manifold we want to be able to compute derivatives, meaning we should be able to define a tangent plane (after all we will recover the Hamiltonian formalism and the cotangent bundle).

So now we are ready to begin the journey of deriving the two products we talked about last time. It will turn out that the symmetric product describes the ontology and the skew-symmetric one the dynamic. Their compatibility condition (which will be derived too) will help recover the quantum and classical mechanics. All this will come out of the two new invariance laws stated above!!! If you want to follow along in a technical paper (and peek into the future post contents) use http://arxiv.org/abs/1303.3935 as guidance. Let us begin…

First we consider a set of (abstract) unspecified operations {o} which include local laws of nature. As concrete examples of such operations in quantum mechanics, we think of the commutator understood as a product, and the Jordan product. At this point we do not assume any properties for those products, or even that they should exist. How can we mathematically state that the set {o} remains invariant under time evolution? The idea is that of an isomorphism: “o“ at time t must be isomorphic with “o” at time t+delta t because the isomorphism preserves the algebraic relationships. If T is a time translation operator (T A(t) = A(t+ delta t)) the isomorphism can be written as:

T [G o H] = [T(G) o T(H)]

Or equivalently:

(G  o H)(t +delta t) = G (t +delta t) o H(t+delta t)

This is the only consequence we can derive from the invariance of the laws of nature under time evolution.

Now consider infinitesimal time evolutions and use the ability to derivate. This introduces a tangent plane at “t” and a (particular) vector field associated with (a particular) time evolution.

If T_epsilon is a particular infinitesimal (time) translation operator we have:

T_epsilon [G(t) o H(t)] = [T_epsilon G(t) o T_epsilon H(t)]

If rho is one of the products in the set {o} corresponding to the time translation transformation T_epsilon, we can construct the following product between a distinguished f and any g:

f ρ g = [T_epsilon g – fg]/ ε in the limit of ε -> 0 (we pick this as the definition)

equivalently:

f (I + ε ρ) g = T_epsilon g

(Here T_epsilon and “f” are not arbitrary, but depend on each other. “f” in general corresponds to a particular Hamiltonian, and ρ corresponds to the Poisson bracket in classical mechanics and the commutator in quantum mechanics.)

T_epsilon can be uniquely represented as f (I + ε ρ) because if f (I + ε ρ) = T_epsilon = r (I + ε η), in the zero order: f=r. In turn this means that f (I + ε ρ) = T_epsilon = f (I + ε η) and therefore: ρ = η.

We generalize the product ρ for all f’s and g’s by repeating the argument for all conceivable dynamics. To make sure the domains of f and g are identical and well behaved, in case of pathologies, we can restrict the domain of g’s to the span of all possible f’s.

From the invariance of the laws of Nature under time evolution we have:

f (I + ε ρ) (g o h) = T_epsilon (g o h) = (T_epsilon g) o (T_epsilon h) =

= [f (I + ε ρ) g] o [f (I + ε ρ) h]

In first order in epsilon we have a left Leibnitz identity:

f ρ (g o h) = (f ρ g) o h + g o (f ρ h)

for any product “o” in the set {o} and for any f, g, h.

Similarly one derives a right Leibnitz identity:

(g o h) ρ f = (g ρ f) o h + g o (h ρ f)

The Leibnitz identities turn out to be of critical importance as we will see in subsequent posts.

In summary, the fact that the laws of nature do not change during time evolution along with differentiation demands the existence of left and right Leibnitz identities for an abstract product ρ.

As an important note, we do not know at this time that ρ is necessarily skew-symmetric. If we assume this from the very beginning the whole derivation is much shorter.

Next time we’ll show that dynamic alone is not enough and we will need a supporting product which in the end will show that it corresponds to observables (ontology). Then we’ll establish the most general relations between dynamic and ontology.

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