## Why are unitarity violations fatal for quantum mechanics?

Last post attracted a lot of attention and criticism from Lubos Motl and I will postponed the scheduled presentation of this week and instead try to address that criticism head on. When discussing quantum mechanics one can take three points of view: physical, mathematical, and philosophical. In the past I had an argument with Lubos on Boolean vs. quantum logic in quantum mechanics, and the root cause of the disagreement then was on the mathematical vs. the physical points of view. This time the root cause of disagreement is due to the physical vs. philosophical approach. Lubos sees no value in the quantum foundations community because the proper interpretation was settled in his opinion long time ago and all quantum foundations practitioners must be crackpots (obviously there is no love lost between the quantum foundation community and Lubos). Today I will not take the philosophical point of view and instead I will attack the argument from the mathematical side and attempt to show that the measurement problem is still an open problem (not only Lubos but also some members of the foundation community disagree with this too).

The problem starts from the category theory approach in quantum mechanics reconstruction. The mathematical formalism of quantum mechanics can be derived from two physical principles:
• invariance of the laws of nature under time evolution
• invariance of the laws of nature under physical system composition
Those two physical principles are easy to understand: invariance under time evolution means that the laws of nature are the same today as they were yesterday, or they will be tomorrow. Invariance under composition means that the laws of nature do not change with additional degrees of freedom. How can you derive the formalism of quantum mechanics (Hilbert spaces, self adjoint operators, etc) from those physical principles? With the help of the algebraic formulation of quantum mechanics.There a C* algebra can be decomposed in a symmetric product (the Jordan algebra of observables) and a skew-symmetric product (the Lie algebra of generators). Since Hilbert space composition is done using the tensor product and since there is a universal property of tensor products linking products with tensor products, imposing the physical principle of invariance under composition has deep algebraic consequences. It can be shown that those algebraic consequences completely determine the C* algebra formulation of quantum mechanics and to recover the full formalism then one has to apply the GNS construction. But to prove all this you need a starting point, and that is the Leibniz identity which comes from the invariance of the laws of nature under time evolution. Violate Leibniz identity and the whole mathematical formalism of quantum mechanics becomes mathematically inconsistent.

Why is this important? Because as it is well known by the mathematics community, the Leibniz identity in the algebraic formulation corresponds to unitarity in the usual formulation. No unitarity during collapse means no Leibniz identity which means mathematical disaster. This is why I am seeking a unitary solution to the collapse problem. This is not a fool's errand, as category theory which highlighted the problem in the first place also points to the unique way to solve it using the Grothendieck group construction. In the process the quantum mechanics interpretation which emerges is that of Copenhagen. And this follows as a necessary mathematical result and not as a crusade against other interpretations.

Since the deep relationship between Leibniz identity and unitarity is not well known outside the mathematical physical community, I will now present it below:

#### Unitarity from Leibniz:

This follows from the categorical approach of quantum mechanics reconstruction. I presented this in a series of posts starting with this one.

#### Leibniz from unitarity:

To see the inverse implication we need to go in depth in the Jordan-Lie algebraic formulation of quantum mechanics. The algebraic formalism is a unifying framework for both quantum and classical mechanics and the only difference is that the observables bipartite product $$\sigma_{12}$$ has an additional element in quantum mechanics: $$-\alpha_1 \otimes \alpha_2$$. This additional element prevents the Bell locality factorization  (the observable bipartite product $$\sigma_{12}$$ can no longer be factorized in terms of $$\sigma_1$$ and $$\sigma_2$$ and this prevents in turn the state factorization) and makes possible superposition and continuous transitions between pure states.

Classical mechanics is described by a Poisson algebra while quantum mechanics is described by a Jordan-Lie algebra. If we add norm properties to a Jordan-Lie algebra, we get a J L B (Jordan, Lie, Banach) algebra which is the real part of a C* algebra. The C* algebra is the algebra of bounded operators on some Hilbert space arising out of GNS construction. The pure state space $$\cal{P}(\mathfrak{A})$$ of a C* algebra $$\mathfrak{A}$$ is a Poisson space with transition probability. Unitarity means that the Hamiltonian flow of the states generated by a given observable $$\psi\mapsto \psi(t)$$ preserves the transition probability $$p$$:

$$p(\psi(t), \phi(t)) = p(\psi, \phi)$$

This definition is more general than the usual definition of unitarity in a Hilbert space, because it works for both the Hilbert space representation and for the state space of the C* algebra. The definition also applies to the classical case. If the classical case there is no superposition and the transition probabilities are trivial: $$p(\psi, \phi) = \delta_{\psi \phi}$$.

To derive Leibniz identity from unitarity one can proceed in two steps. First the algebra of observables $$\mathfrak{A}_{\mathbb R}$$ is recovered from the pure state space. Then the Hamiltonian flow $$\psi\mapsto \psi(t)$$ defines a Jordan homomorphism and the derivative with respect to time of the homomorphism property yields the Leibniz identity.

Given a transition probability space we can define linear combinations of transition probabilities and this defines a real vector space $$\mathfrak{A}_{\mathbb R} (\mathcal{P})$$. In the quantum mechanics case the elements of this vector space  have a spectral decomposition $$A = \sum_j \lambda_j p_{e_j}$$ which allows the definition of a squaring map: $$A^2 = \sum_j \lambda^2_j p_{e_j}$$. This in turn is used to define the Jordan product:

$$A\sigma B = \frac{1}{4} ({(A+B)}^2 - {(A-B)}^2 )$$

With the sup norm and the product $$\sigma$$, $$\mathfrak{A}_{\mathbb R} (\mathcal{P})$$ now becomes a J B (Jordan Banach) algebra and the first step is complete: starting from the pure state space equipped with transition probability we arrived at the Jordan algebra of observables. In the classical mechanics case the Jordan product is simply the regular function multiplication.

For the second step, we use the Poisson structure and the Hamiltonian flow: $$\psi\mapsto \psi(t)$$. For each element $$A$$ of $$\mathfrak{A}_{\mathbb R} (\mathcal{P})$$ (corresponding to an operator in a Hilbert space by the GNS construction) we can define a one-parameter map $$\beta_t$$ given by $$\beta_t (A): \psi \mapsto A(\psi (t))$$. Then we have: $$\beta_t (A \sigma B) = \beta_t(A) \sigma \beta_t(B)$$ because $$\beta_t(A^2) = \beta_t(A)^2$$. As such $$\beta_t$$ is a Jordan homomorphism.

From the Hamiltonian flow we have:

$$\frac{{\rm d} A}{{\rm d} t}(\psi (t)) = \{h, A\}(\psi (t))$$

We take the time derivative of the isomorphism $$\beta_t (A \sigma B) = \beta_t(A) \sigma \beta_t(B)$$ and we obtain the Leibniz identity.

We need to point a notation difference between what we and the literature calls Leibniz rule. In  the mathematical literature the Leibniz identity defines only how the Poisson bracket $$\alpha$$ acts on the observable algebra $$\sigma$$ and the proof was outlined above. In my case Leibniz identity applies to all algebraic products. However the only other algebraic product is the Poisson bracket itself and because its skew-symmetry the Leibniz identity becomes the Jacobi identity. The Jacobi identity is assumed by the structure of a Poisson space and we do not need to derive it from unitarity. Both classical and quantum mechanics have a Poisson space structure.

Now back to the philosophical point of view. Various quantum mechanics interpretations are nothing but distinct frameworks for solving the measurement problem. The basic problem is that given a unitary explanation of one outcome A by introducing a particular coupling between the measurement device and the quantum system, and given a  unitary explanation of another outcome B, then by superposition you get a superposition of outcomes and this is nonsense (there are no dead and alive cats). Now which framework for solving the measurement problem is the correct one? Because the measurement problem must explain the non-unitary collapse, and since non-unitarity makes the mathematical framework of quantum mechanics inconsistent, the mathematical solution ultimately points out the right interpretation. So far everything in the category theory approach points towards the Copenhagen family of interpretation as the correct explanation.

If I can offer an analogy, the Grothendieck approach for solving the measurement problem without spoiling unitarity is to quantum mechanics what the Higgs mechanics is to field theory (introducing mass without spoiling the gauge invariance). The problem is still open and it is work in progress. The basic idea is that instead of one Hilbert space describing the collapse there are actually many Hilbert spaces linked by an equivalence relationship.The equivalence relationship encodes mathematically the physical principle of outcome randomness in quantum mechanics (this comes in turn from operator non-commutativity). Each Hilbert space corresponds to a GNS representation of a C* algebra state corresponding to a potential outcome. The one and only outcome is realized when the equivalence relationship is spontaneously broken by a pure unitary mechanism. The observer (measurement device) plays an essential role and the result of measurement does not exist independent of measurement. However, the consciousness of the observer plays no role whatsoever. There is no need for the measurement device to be described classically.

1. It's hopeless, Florin. I wrote a reply on my blog.

1. If all is so old and trivial, what is the relationship between Lie and Jordan algebras in QM and what does it correspond to from the physical point of view?

2. When the dust settled, it became clear that nothing like the Jordan algebras - or other non-associative algebras - are needed in quantum mechanics, certainly not for the foundations and basic definitions that are needed to do physics - pretty much any physics. That's why we are not teaching Jordan algebras in most QM courses.

Lie algebras are all closed algebras of operators under the commutators, so they're everywhere in quantum mechanics and describe the space of all possible observables, or its subset. Every undergraduate sophomore has been exposed to them in basic courses of QM. They're most typically used for the restricted Lie algebras of symmetries, such as the SU(2) or SO(3) symmetries under rotations.

All these things are extremely old, even the Jordan algebras that Jordan thought would be very helpful but they were not. Leibniz lived in 1646-1716, all the Lie, Jordan, Killing, Cartan stuff is from the 19th century mathematics, and is taught in first two semesters of undergraduate linear algebra. Matrices etc. were simple enough so that Heisenberg reinvented them when he needed to invent quantum mechanics.

It is totally foolish to create a fog of mystery about any of these things in 2016 - just like to doubt heliocentrism.

And concerning the topic in the title, it is not true that the collapse means any non-unitarity. Unitarity is only required - for consistency - for the probability amplitudes of evolution, i.e. the pre-collapse complex numbers, and those are exactly unitary in all theories we use.

It makes no sense to ask whether the change of the wave function "with collapse" is unitary because it is not even a map - there isn't a well-defined final state f(v) for a given initial state v (because of the randomness of quantum mechanics). So it's just silly to talk about the unitarity "after the collapse". Unitarity is a property Yes/No that may only hold or not hold for maps.

Even if one defined some sense in which the collapse is a "non-unitarity" of the evolution, even though this evolution-with-the-collapse isn't even given by a map, it would represent no problem whatever. Only non-unitarity of the pre-collapse probability amplitudes is a problem. It's clear that you have just superficially heard that non-unitarity is a problem but you didn't bother to ask what unitarity exactly was and what was demanded to be unitary. So you conflated it with some vague misconceptions about the measurement's being "different from the unitary evolution" and you use it to claim that there is a problem. There is no problem here, except in your head.

Your text is plagued by lots of silly mistakes pretty much in every point you are making. For example, you are suggesting that unitarity is kind of equivalent to the Leibniz identity. But it's silly. The Leibniz identity is a mathematical fact that always holds for derivations (in the algebraic sense) - the Lie-algebra-like generators of Lie groups - whether the groups are unitary or not. Unitarity is equivalent to the (anti)Hermiticity of the generators.

And so on. None of these things you do has any relationship with science. In physics, it's at most pseudoscience, and writing this would-be clever but actually stupid stuff belongs to the humanities, on par with essays about the discrimination of female transcendental numbers etc. Junk.

3. Lie, Jordan... should have been Lie, Jacobi.

4. The one-to-one map between the Lie algebra and the Jordan algebra is known in the literature as "dynamic correspondence". This has a deep relationship with Noether theorem. There was a generalization of quantum mechanics by Segal (the S in GNS) based on the relationship between observables and states and this generalization was unphysical because it violated Noether's theorem precisely because it did not respect dynamic correspondence.

Most of what you wrote at your blog and here is plagued by silly mistakes. Leibniz-unitarity relationship is well known by the mathematical physicist community for many years. The proof is purely mathematical and you cannot argue with it just as you cannot argue with 2+2=4.

5. Please, don't assume that your readers are complete idiots. One may open scholar.google.com and search for

"dynamic correspondence" jordan lie

He will only find your 4 papers with this nonsensical term, they have 2,0,1,0 citations, respectively, and one of the three is yours. Joy Christian has surely been a more successful crackpot in selling his junk than you were.

6. Very funny. You are way, way, way outside your league here. Do you know that story of emperor's new clothes when after someone shouts the truth, the emperor continues with the charade to save face?

The literature does not have only 2-3 papers on this, but contains the essential work of von Neuman algebra classifications by Connes (you know: an obscure Fields medal winner who won the prize due to this particular work lol), the Tomita-Takesaki theory, the works of Kadison, Alfsen, Shultz, Stormer, Hanke-Olsen, Araki, etc, etc.

2. This is a funny argument in a way. If one is looking at Hermitian symmetric spaces with H = G/K coset construction, then g = der(G) may be a nonassociative algebra, but the physically relevant elements are the generators of H, h = der(H) that are associative. This is in line with the whole physics = paths or states modulo diffeomorphisms, to put this in ordinary language, and nonassociativity may just exist as a gadget that permits a larger reduction with “modulo nonassociativity.” Nonassociative relationships are then just redundancies that are removed from the relevant physics.

It is clear that we do not want direct nonassociative physics. You have no Jacobi identity and thus no conservation or continuity principles. This does not mean these things can't play an interesting role in the foundations of physics.

LC

1. I don't know if non-associativity plays any role. Octonionic QM is not physical.

2. Nonassociative sets, such as the 7-triplets in the octonions, are quaternions. It is not hard to derive the basic YM gauge theory with quaternion derivative. It is a quaternion generalization of how to derive the Cauchy-Riemann conditions in complex variables. In octonions these 7 quaternion systems are equivalent "modulo nonassociativity." This is a seven fold redundancy, which for complex octonions is 14-fold and has correspondence to G2 holonomy.

Nonassociativity is a way of eliminating gauge redundancies. It is the case there are no physically real nonassociative field equations.

LC

3. Nonassociative sets, such as the 7-triplets in the octonions, are quaternions. It is not hard to derive the basic YM gauge theory with quaternion derivative. It is a quaternion generalization of how to derive the Cauchy-Riemann conditions in complex variables. In octonions these 7 quaternion systems are equivalent "modulo nonassociativity." This is a seven fold redundancy, which for complex octonions is 14-fold and has correspondence to G2 holonomy.

Nonassociativity is a way of eliminating gauge redundancies. It is the case there are no physically real nonassociative field equations.

LC

4. In the quantum case associativity means that if I have 3 experiments A, B, and C such that the output of one is the input (preparation) of the other then it makes no difference where I put the mental cut between them. This is why QM over octonions makes no operational sense. There is also a mathematical reason too but I need a full length presentation to go in detail over it.

Quaternionic QM makes the very same predictions as complex QM in its (limited) domain of validity and Adler has a monograph on it. The one key difference in quaternionic QM is that it does not allow a tensor product and the reason for it is that quaternionic QM is basically a constraint complex QM.

3. As I said, the point of octonions is not that actual quantum measurements have nonassociative results. It is as a way of understanding moduli or the quotient of a group with a divisor of gauge redundancies.

Quaternionic QM, or should it be better put QM of quaterionic fields, is applicable in conformal field theory CFT_4 that is dual to the anti-de Sitter spacetime in 5-dim. The point of doing QM with quaternions is not so much that it is "better QM," but it is applicable with gauge fields that have quaterion representation.

LC

1. Lawrence,

One way to understand quantum mechanics is as a constrained classical mechanics which preserves a metric in addition to a symplectic structure. For this to work the observables cannot be like in classical physics, but they must commute with sqrt(-1). In other words they must be Hermitean.

Now on quaternionic QM, in addition to i=sqrt(-1) you also have j=sqrt(-1) and k=ij=sqrt(-1), and the observables in quanternionic QM must commute with j as well. Physically j corresponds to time reversal and j*j=-1 is true for fermions. Quaternionic QM describes fermionic particles, and real QM describes bosonic particles. Now two fermions do not make another fermion and this is the reason quaternionc QM does not admit a tensor product.

Real and quaternionc QM are constrained QM, and the constrain comes from the behavior under time reversal.

2. I think you mean that i^2 = j^2 = k^2 = ijk = -1.

The constraint you refer to is that quantum mechanical systems have Usp(n) structure, which is in a sense means unitary plus symplectic structure. The Dirac operator is quaternionic, and in greater generality the (½ ,1) superfields or multiplets in supersymmetry are also. Conversely, chiral gauge bosons in (1, 3/2) multiplets have quaternion gauge symmetries.

It is not hard to derive YM gauge field from quaternions in much the same way one derives the Cauchy-Riemann conditions with complex numbers. If one tries to do the same with octonions you end up with nothing particularly interesting, but there is this issue of nonassociativity. As I said quantum systems with exceptional or Freudenthal structure should have physically real fields “mod nonassociativity.” The constraint that comes from octonions is that the physically real field theory is a quotient group or Hermitian space.

The time reversal issue you mention is just an example of the antisymmetric form of fermion wave functions and the Pauli exclusion principle.

LC

4. In a vilage there were two sweets shops. One day both shop keeper had argument and it turned into fight. The started throwing sweets at each other. Villegers were very happy becasue the got some free sweets to eat.