Saturday, October 26, 2013

What is the number system of quantum mechanics?

Quaternionic quantum mechanics

Observables are Hermitean operators. In matrix notation this means that an observable is the same as its transpose and complex conjugate. The natural question to ask is: what mathematical properties can be derived from this observation. Can observables be combined to create other observables? Do observables form an algebra for example?

The answer is yes provided we have a product which preserves the hermiticity. However, a problem comes from transposition: if C = A B , then C^dagger = (A B)^{dagger} = B^{dagger} A^{dagger} and if A = A^{dagger} and B = B^{dagger} then C^{dagger} DOES NOT EQUAL C Can this be fixed? Indeed if we define a symmetrized product called the Jordan product:

A o B = (A B + B A)

This product is not quite associative, but respects a weaker property called power associativity: A^(m+n) = A^m o A^n

This allows to have well defined polynomials and we can also define the product in terms of operators squared:

A o B = ¼ (A + B)^2 – ¼ (A-B)^2

If we also demand that the algebra is in some sense “real”:

A o A + B o B = 0 --> A = B = 0

Then we have what it is called: a “real Jordan algebra”.

Now in quantum mechanics the states are defined only up to a phase because the Born rule: p_A(Psi) = |<Psi| A | Psi>|^2 ignores the phase which is an unobservable quantity.

So if a Hilbert space is about orthogonality, the physical states correspond to rays in a Hilbert space. This means that we are looking at projective spaces.

Do we have a complete classification of real Jordan algebras? Yes, and it is as follows:

  • Jordan algebra over reals with dimension N(N+1)/2 corresponding to RP^(N-1) – real projective space
  • Jordan algebra over complex numbers with dimension N^2 corresponding to CP^(N-1) – complex projective space
  • Jordan algebra over quaternions with dimension N(2N-1) corresponding to HP^(N-1) – quaternionic projective space
  • Spin factors over SO(n,1) with dimension n+1 corresponding to S^{n-1}, the n-1 sphere
  • Albert octonionic algebra with dimension 27 corresponding to OP^2 – octonionic projective space in 2 dimensions.

If you compare the dimensionality of the projective spaces above with the number of linear independent projectors from the prior posts you’ll see that they are the same (and for very good reason).

From the Jordan algebra point of view one can see that a quaternionic quantum mechanics is in principle possible, but what does it mean and what it is gained by that?

Fortunately there is an outstanding resource available, a comprehensive monograph by Stephen Adler:  “Quaternionic Quantum Mechanics and Quantum Fields

Adler starts from the fundamental definition of a quantum system: “in quantum mechanics, probability amplitudes, rather than probabilities, superimpose

From this and Born rule one demands the existence of a “modulus function” N with the following properties:

            N(0) = 0
            N(phi) > 0 if phi is not 0
            N(r phi) = |r| N(phi) with r real
            N(phi_1 + phi_2) <= N(phi_1) + N(phi_2)

By a mathematical theorem, the properties above restrict the number system for quantum mechanics to be: reals, complex numbers, quaternions, and octonions). In later posts we shall see that this is not the complete classification and there are many more number systems possible.

In quaternionic quantum mechanics, physical states are in one-to-one correspondence with unit rays in the quaternionic Hilbert space of the form:

| f > = { |f omega > : |omega| = 1}

What does this mean? Quaternions correspond to rotations and rotations are non-commutative. An easy way to see this at home is to rotate a book on two axis:

But if quantions are non-commutative, we must be very careful about the order of terms. In particular there are distinct left and right eigenvalue problems!!!

So why bother with this order hassle which is non existent for complex quantum mechanics? Because it may provide new physics, new insight, and it may lead to unification in terms of fundamental forces and/or relativity.

For example, rotation in 3D correspond to SO(3) which is isomorphic with SU(2) (the dimensionality of SU(2) = 2^2 -1 = 3 = dimensionality of SO(3)) and the unit rays in the quaternionic Hilbert space are defined up to an internal degree of freedom of SU(2). And SU(2) is the gauge group of the weak force so we may naturally understand the weak force in this formalism. It turns out that quaternionic quantum mechanics does not make any new predictions which are different than the predictions of complex quantum mechanics, but quaternionic quantum mechanics is a constrained complex quantum mechanics.

Here is how the story goes:

Observables are maps which transform a wavefunction in another wavefunction. By Riesz representation theorem on Hilbert spaces, observables can transform two wavefunctions into real number (this is because of Born rule).

As operator algebra, the observables obey a Poisson bracket algebra isomorphic with a commutator algebra. In this classical mechanics picture, to quantify and arrive at quantum mechanics we need to impose a constraint to preserve hermiticity. The constraint is: observables must commute with the imaginary number. Complex quantum mechanics is classical mechanics on CP^n formulated as a constrained Hamiltonian system on C^n. The imaginary unit realizes another observables duality called the dynamic correspondence which maps observables to generators, or the map between a Jordan algebra to a Lie algebra.

Suppose now we have a system with time symmetry Theta. By Wigner’s theorem an isometry Theta is implemented unitary or anti-unitary:

<psi Theta | Theta phi> = <psi | phi>

which demands Theta^2 = +/- 1

The sign choice depends whether the spin is integer (bosons) or half-integer (fermions).

For fermionic physics, preserving the statistics means that the observables obey a superselection rule:

[O, Theta] = 0

We knew from above that this holds as well:

[O, J] = 0 with J the imaginary unit

So we can collect the terms now and define i, j, k:

i = J, j = Theta, k = J Theta

with the following properties:
i^2 = j^2 = k^2 = -1, ij = k, jk = i, ki = j

This defines the algebra of quaternions!!!

Quaternionic quantum mechanics is the quantum mechanics of fermions!!!

Now it is clear why there is no tensor product in quaternionic quantum mechanics: two fermions cannot be understood as another fermion!

Similarly, real quantum mechanics can be associated with bosons and there is no tensor product there also for the same reason. (As a strong word of caution, the association with bosons and fermions is not one-to-one.)

Next time I’ll discuss another quantum mechanics number system called “quantionic quantum mechanics” which unifies quantum mechanics and relativity!!!


  1. It is interesting that you conclude that quaternionic QM is the fermion. This is of course the case, but where the Dirac matrices have a reducible representation, and the system is equivalent to a pair of spinor theories. The matrix theory is Cl(1, 3, R), which includes quaternions as well as hypercomplex number systems. The Dirac matrices are reducible quaternions, or in effect hypercomplex system of spinors.

    It is possible to consider the situation where the Dirac equation for the massless field is of the form

    i∂^μ(γ_μψ) = 0 = iγ_μ∂^ μψ + i(∂^μ γ_μ)ψ.

    Now we multiply the whole equation byγ^ν by

    iγ_ν ∂^μ(γ_μψ) = 0 = iγ_νγ_μ∂^ μψ + iγ_ν (∂^μγ_μ)ψ

    and use γ_μγ_ν = 4g_{μν} so the differential of the gamma matrix is

    γ_ν (∂^μγ_μ) = 4∂^μg_{μν} – γ_μ (∂^μγ_ν)

    so by the symmetry of the indices μ and ν and a little gamma algebra

    iγ_μ∂^ μψ + 2iγ^ν∂^μg_{μν}ψ = 0.

    Now of course we use the definition of the connection coefficient to give

    γ^ν∂^μg_{μν} = γ^νg^{μα}∂_αμg_{μν} = γ^νg^{μα}(Γ^β_{αν}g_{βμ} + Γ^β_{αμ}g_{βν})

    = γ^νΓ^β_{βν} + γ_βΓ^β_{αμ}g^{μα}.

    This then gives a spacetime gauge covariant form of the Dirac equation.

    The gauge covariant quantum operator may then have elements that are quaterionions of a gauge field. A gauge field may be expressed according to a Clifford basis. In the case I illustrate here gravity may also emerge from Dirac theory if the representation of the Dirac matrices is local and has a chart/atlas construction which defines connections and curvatures.

    Cheers LC

  2. Lawrence,

    Please stay tuned for the next posts where I will touch on Dirac's equation heavily. Quaternionic QM turned out to be a disappointment from the unification or the new physics point of view. However it is a step in the right direction. The next number system: quantions contain quaternions as a subalgebra and they do unify QM and relativity. In the process one recovers Dirac's equation, the degrees of freedom for spin, and another factorization of the d'Alembertian. In fact quantions are intimately linked with spinors (there is a one-to-one correspondence).

    The problem with quaternionic QM is that any of the 3 imaginary elements: i,j,k can play the role of the map between observables and generators (or the role of i in complex QM). Once you pick any one of the 3 to distinguish what one calls hermitean, all quaternionic QM predictions become the standard complex QM predictions.

    There was a paper on the archive about quaternionic QM claiming one can get over Tsirelson's bound, but from the post it is clear that this is wrong: quaternionic QM is complex QM with a constraint, and it can achieve less or equal correlations, not more.

    To get to CPT and Dirac's equation one needs to go to second quantization (and quantions are about second quantization). One may even call string theory 3rd quantization. I'll show how this works: it is related with how one can arrive at probability predictions.

  3. I finished reading “Quantum Theory From Five Reasonable Axioms,” Lucien Hardy. The paper was interesting, though maybe a bit pedantic at times. I agree with the general thrust of the paper, or the conclusion that K = N^2. The fermions have a partitioned space with states ψ_A and ψ_A’ so there are two wave-field equations

    ∇^{AA’}ψ_A = imψ^A’, ∇^{AA’}ψ_A’ = imψ^A

    where there is some normalization factor I absorbed into m. These two spinor field equations, Weyl equations, comprise the Dirac equation. This means the quaternions have an irreducible representation, which as a split matrix still gives K = N^2. The difference is that N --- > 2N for the two components of the field.

    What is curious is the Majorana neutrino. The charge conjugation operation means that the full spinor equation is not so irreducible. The Majorana neutrino cold be the form that sterile neutrinos take.

    Cheers LC

  4. "Quantum Theory From Five Reasonable Axioms" is now a famous paper. Its major weakness is on stopping on N^2 and not clarifying N^3 and higher.

    Grgin's quantionic papers and books are much more interesting and deeper, but they are not known. I'll promise some amazing fireworks in the next posts. It is a shame Grgin's work is not well known. This work puts Dirac's equation in a whole new paradigm.

  5. In studying the Hardy paper it occurred to me that the second proof can be made into a statement about spacetime if the continuum is equivalent to a condition for Fubini’s theorem. The causal condition for geodesics and null paths are then a statement of the continuum axiom, and sequences of these should give horizon conditions.

    I seem to remember a bit about quantions. As I recall these has an SO(4,2) or ~ SU(2,2) symmetry which is the isometry group for AdS_5. The AdS_5 ~ SO(4,2)/SO(4,1) as a quotient group and a conformal compact condition on the AdS is given by the further quotient with a discrete group. These Kleinian group or quotient manifolds are interesting structures.

    Cheers LC

  6. Quantions came from SO(4,2), but I don't think this is significant (I don't think this anymore - I used to think that). The reason is that it is not pure SO(4,2), but SO(4,2) with a superselection rule which gives rise to the quantization procedure via a mechanism called "internal complexification". What this means is that one element of SO(4,2) plays the role of sqrt(-1) and then we look at a sub-algebra of SO(4,2) subject to a constraint. In the process the original SO(4,2) vanishes.

    Quantions are related to SL(2,C), in fact they are a direct sum of 2 copies of SL(2,C) and the hermitean elements correspond to SO(3,1). The Jordan algebra in this case is a spin factor and Hardy's formula is K = 4N^2 The additional factor 4 gives rise to the spin and a quantionic-spinor duality. Initially I thought the formula was K=N^3 until I computed explicitly the projector basis.

    1. The internal complexification is due to the pseudocomplex nature of SU(2,2) ~ SO(4,2). The signature of this group [+,+,-,-] has correlations to a symplectic structure of a pseudocomplex system.

      The AdS_n ~ CFT_{n-1} implies that the AdS_5 = SO(4,2)/SO(4,1) in the interior AdS_5 is a set of gravitational symmetries equivalent to the conformal field theory on the boundary of the AdS_5. This is a Clifford Cl(3,1) or an SO(4) gauge theory. SO(4) is a Euclideanized theory of gravity, while CL(3,1) is the structure of the Dirac matrices. As I indicated the other day general relativity can be derived using the Dirac matrices.

      Cheers LC

  7. In internal complexification one picks a bivector of signature (+1,-1) and selects the elements of SO(4,2) which commutes with the bivector. The resulting algebra is the quantionic algebra, the simplest von Neumann type 1 factor. Quantions satisfy the bicommutant theorem and the center is trivial. Any von Neumann type 1 factor can be made into a C* Hilbert module.

    The next factor is:

    147 000 000
    258 000 000
    369 000 000
    000 147 000
    000 258 000
    000 369 000
    000 000 147
    000 000 258
    000 000 369

    and this gives rise to U(1)xSU(3) and so on and so forth for the U(1)xSU(n) gauge theory.

    But only for U(1)xSU(2) one can interpret this as current probability. If the 2 block diagonal algebra is over reals or quaternions, then the hermitean elements are isomorphic with SO(2,1) or SO(5,1)