Saturday, October 12, 2013

What is the number system of quantum mechanics?

Quantum mechanics from 5 reasonable axioms

I will now start a new series about the number system of quantum mechanics. There are only 4 possible division number systems: real numbers, complex numbers, quaternions, and octonions. Since there is a deep connection between them, projective geometry, and Jordan algebras (an excellent reference is Baez’s Octonions paper), it is expected that quantum mechanics can be formulated in Hilbert spaces over those number systems.  Also we may ask if there are other possible number systems for quantum mechanics (and the surprising answer as we will show in this series is yes).

From the start one can eliminate the octonions because they lack associativity and the tensor product is associative As division number systems, there is a mathematical progression called the Cayley-Dickson construction which builds complex numbers out of real numbers, quaternions out of complex numbers and so on, and at each step the resulting number system loses some property. That is why anything beyond quaternions is not interesting for quantum mechanics.

If we demand the possibility of the tensor product, then we are left only with complex numbers, as a nice argument by Lucien Hardy shows it in .

Hardy’s argument is instrumentalist, but at core this result stems from dimensional analysis for Lie groups as Jochen Rau showed in

Following Hardy, let us introduce two numbers K and N:

  • The number of degrees of freedom, K, is defined as the minimum number of probability measurements needed to determine the state, or, more roughly, as the number of real parameters required to specify the state.
  • The dimension, N, is defined as the maximum number of states that can be reliably distinguished from one another in a single shot measurement.

From composability, for two subsystems A and B, N = N_a N_b and K = K_a K_b and assuming that K depends on N: K=K(N), Hardy shows that the following relationship holds:

K = N^r with r=1,2,3,…

r=1 corresponds to classical mechanics, and r=2 corresponds to quantum mechanics over complex numbers.

r>1 means that the whole system A+B has more information than each of its parts, and this is a characteristic of quantum mechanics and its peculiar correlations which go above the Bell local realism limit.

It is very instructive to go deeper and see how N^2 is constructed in terms of projector operators. (In later posts we shall see that Hardy’s relation is not the most general possible by a constructive counterexample giving the explicit list of projectors.)

For ordinary quantum mechanics over complex numbers, in the case of finite dimensionality with the Hilbert space base given by the following ket vectors: |1>, |2>,…,|n> there is the following projector basis:

N projectors of the form: |j><j|
½ N(N-1) projectors of the form: |mn><mn|
½ N(N-1) projectors of the form: |MN><MN|

|mn> = 1/sqrt(2) (|m> + |n> )
            |MN> = = 1/sqrt(2) (|m> + i |n> )

Now N^2 = N*1 + ½ N (N-1) *2

In general this relationship holds:

K = N a + 1/2! N (N-1) b + 1/3! N (N-1) (N-2) c + …

The “a” terms correspond to a local realistic interpretation of the wavefunction, the “b” terms correspond to the new physics of quantum superposition and the higher terms correspond to r > 2 theories where more than two wavefunctions can be combined in such a way that they produce new physics beyond superposition (I’ll present a possible interpretation in subsequent posts).

The reason for the two ½ N(N-1) terms in complex quantum mechanics is the fact that complex numbers can be understood as a pair of real numbers. So what can we say about real or quaternionic quantum mechanics?

Or real quantum mechanics we can only construct |mn> and we have:

K = N + 1/2! N(N-1) = N^2 / 2 +N/2, not of the form N^r

On quaternionic quantum mechanics we can construct 4 such elements using 1,i,j,k:

|mn>_{1ijk} = 1/sqrt(2) (|m> + {1ijk}|n>) and

K = N + 1/2! N(N-1) *4 = 2 N^2 -N, also not of the form N^r

Real and quaternionic quantum mechanics lacks the ability to form tensor products and we’ll later explain why this is so. For now I only want to point out that the failure to satisfy K = K_a K_b implies the failure to satisfy De Finetti theorem which relates to quantum state tomography


  1. I think few people have said that QM is quaternionic or octonionic. The closest QM get to being quaterionic is with the Dirac equation, but in the standard representation it is reducible to spinors, which are two Weyl equations. If this holds in the standard representation it means it ultimately holds in any representation of the Dirac matrices. Thus QM appears safely within the domain of C. Quaternions play a role with gauge theory however, where gauge connections A^a_μ obey commutation rules such as

    [A^a_μ, A^b_ν] =ε_{μνρ}T^{abc}A^{cρ},

    that can be expressed according to quaternions. These are the generators of unitary operators U = exp(q^a∫A^a_μ dx^μ) which can transform quantum states as ψ --- > Uψ.

    The state vectors are on the group level, or better put the transformation rules for state vectors are on the group level. The generators of the transformations can well enough be quaternions. With quantum gravity we really do not know for certain what the rules are. However, gauge/gravity duality with AdS/CFT correspondence does indicate that gauge theory is in effect the “boundary of gravity.” For certain reasons I think this could imply a role for octonions with quantum gravity.

    I do sort of enjoy these, for you are illustrating an area of physics that I have largely a surface knowledge of.

    Cheers LC

  2. I'll dig a bit in the quaternionic case in the next posts and I'l make the link with gauge theory. Octonions were thought for a while to be useful to describe the strong force SU(3), but it turned out to be a red herring. Advances in Jordan algebras proved that Albert algebra were a terminal exceptional case and there are no other algebras possible.

  3. The Albert algebra or Jordan algebra J^3(O) is a system of three octonions in a matrix system. The octonions are one vector plus two spinor forms O_v, O_s, O_s where the two O_s can be conjugates with respect to SUSY generators. This in a way forms a sort of Feynman diagram that has connections to the Dynkin diagram for SO(8) ~ D_4. This system is interesting for if we think of the octonions as a set of vectors that are degenerate under a superselection rule then the system describes the CS Lagrangian. The three octonions give the Leech lattice that is a 24 dimensional subspace of the 26 dimensional Lorentz space related to the bosonic string. This is also found with the Freudenthal determinant. This turns out to be an automorphism of the Conway-Fischer-Greiss system of the monster group.

    The G_2 group is the automorphism of E_8, the group of the octonions. G_2 decomposes into SU(3) + 3 + 3-bar. This is probably the basis for your statement about the E_8 as the group for QCD. I never heard that the whole E_8 group was considered as the primary group for QCD. Garret Lisi has work on how E_8 would gives supergravity. However, he does this in violation of the Coleman-Mandula theorem. Supersymmetry is required to get around that.

    The higher level division algebras H and O only play a role with the generators of unitary groups, for U1 = exp(iA) and U2 = exp(iB) we have that

    ln(U1*U2) = i(A + B) – [A, B] + (i/12)([B, [A, B]] – [A, [A, B]]) + …

    The case where A and B are gauge potentials or integrations on gauge one-forms leads to a quaternion formulism of gauge theory. The third order element may be considered to have nonassociative properties, or for products of three such groups the second term will involve [A, [B, C]] terms that may have nonassociative properties.

    The coset constructions G_3/H_3 = E_{8(8)}/SO*(16), SO*(16) = SO(16,C) ~ SO(32), G_4/H_4 = E_7(7)}/SU(8) are a primiary interest. The algebras g and h are such that g = h + k and that the orbit space of h is equivalent to that of g if h is the algebra for H the maximal compact subgroup. The number refers to 3 and 4 forms, and the G_4/H_4 is the boundary of the G_3/H_3. This coset construction describes the quantum information or qubit space. The E_{8(8)} describes gravitation while the E_{7(7)} is a gauge theory that lies on the boundary. By extension we have then some description possible where an octonionic quantum gravity could describe on the boundary of the space of degrees of freedom a gauge theory that is quaternionic. E_7 decomposes into E_6, which is a particularly nice gauge group for it contains SO(10).

    It is fun to think of these things. There is a prospect for nonassociativity with quantum gravity due to the quantum fluctuations of an event horizon. Three fields, one exterior to a black hole, one on the stretched horizon or just above the horizon and another interior will group together depending upon whether the field near the horizon fluctuates into the exterior or interior.


  4. About octonions and SU(3) I am aware of some 70s attempts by two Hungarian physicists:

    M. Günaydin and F. Gürsey, “Quark structure and octonions”, J. Math. Phys., 14 (1973)
    M. Günaydin and F. Gürsey, “Quark statistics and octonions”, Phys. Rev., D9 (1974)

    To my knowledge those efforts did not pan out. Again, the problem was lack of associativity which makes computations and predictions dependent on the order of operations.

  5. Thanks for the references. These are fairly dated papers. I suspect they are related to work around that same time that attempted to use G_2 as the symmetry of a QCD gauge theory. That did not work out. G_2 and its decomposition as SU(3) on the algebra level leaves su(3) ⊕ 3 ⊕ 3-bar, which has connections to the Eguchi-Hanson metric. G_2 is the automorphism subgroup of E_8 and that might have motivated people to think about octonions as the gauge group for QCD.

    The only physically motivation for nonassociativity is the occurrence of event horizons. Even horizons have an ordering of operation interpretation with fields. Outside of gravitation there would be no reason to entertain any idea about there being nonassociativity. With gravity I think the wave functions are still unitary (mostly) and the quaternion and octonion stats come in with the fields or generators of the wave functions. I say the wave functions are mostly unitary because the occurrence of singularities is a pole that makes functions meromorphic.

    In the Schwarzschile metric perform the substitution 1 – 2m/r = e^u. then

    ds^2 = e^udt^2 – e^{-u)dr^2 + r^2dΩ^2.

    We now have to get dr from

    dr = -2me^u/(1 – e^u)^2du.

    Now the metric is

    ds^2 = e^udt^2 + -4m[e^u/(1 – e^u)^4]du^2 + r^2dΩ^2,

    where I will ignore the angular part for now. The singularity is at u = ∞, where the dt term blows up, and the horizon coordinate singularity at u = 0 is obvious in the du term. My rational was that the singularity had been removed “to infinity” in these coordinates and were then not a direct problem. A wave function in this space will then be unitary everywhere except this point. This is a meromorphic function, which is analytic everywhere but a single pole that can be removed to infinity. The sorts of functions which are transformed by the E8 lattice are Jacobi θ-theta functions, which are meromorphic functions. The removal of the singularity to infinity means the physical impact of the pole is not relevant, or just a residue that is an additive constant. The complex nature of quantum waves is the same, except for a measure ε part of the L^2 measure.

    I suspect this might be a part of the so called firewall problem with the holographic principle and black holes. If I am right, which of course is problematic, then I think unitarity is restored to quantum holography “modulo actions” which have a measure zero impact on the quantum physics.

    I would be curious to see what motivated M. Günaydin and F. Gürsey to think that QCD is connected to the octonions. Maybe there is something there with respect to the gravity/gauge theory duality. I have encountered these authors before and they are pretty decent.

    Cheers LC

  6. Lawrence,

    I think the original motivation was simply the SU(3) symmetry. In the subsequent posts I'll show how to naturally get to the electroweak gauge group. The theory is simply beautiful and goes again to show the brilliant work of Grgin. I hope I'll do the theory some justice on how I'll present it.

    Regarding a black hole, I have no idea what one might mean by a tensor product of two Hilbert spaces one inside and one outside. I mean this from both the instrumentalist and the mathematical point of view. Sure, there is this double story of the far away observer, and the infalling one...

    I think one of the best approaches to solving the black hole information paradox came from Maldacena which thought of the singularity as a boundary condition. People shot holes into the argument, but did not follow it closely to decide who is right and who is wrong.

    QFT near the horizon is very tricky business and I have no good intuition about it.

  7. ...One more comment. On the mathematical difficulties of QM inside a black hole, I am thinking in terms of the path integral formulation and the fact that the paths must go through the singularity too. What this does to QFT is very unclear to me.

  8. The Maldacena idea is related to the AdS/CFT result, but with the black hole. The duality between the interior and exterior fields, or those holographic projected from the horizon to the exterior and those with the singularity according to an interior observer, makes the singularity a boundary. If quantum states are meromorphic, or unitary modulo a finite number of poles at the singularities, the residues associated with the poles carry this information. The fields have an STU duality, where the field according to the exterior observer have a Wilson loop description and those on the interior are an S-dual field configuration with the ‘t Hooft loop space.

    Understanding fields associated with black holes is difficult. There is this problem with the firewall. This stems from the quantum monogamy of quantum states. The entanglement between states near the horizon, inside and outside fields that are entangled can’t remain entangled under a “swap” of entanglement with fields just outside the horizon (the stretched horizon) and Hawking radiation. This means the dimension of the Hilbert space for the two parts (in and out) dim(H_i) = e^{S_i}for i = 1,2 is dim(H_1⊗H_2) = e^{S_1+S_2} ~ e^{2S}is a part of a general space with four subspaces such that the total Hilbert space has e^{4S} total dimension. The four parts being in + out near the horizon, the singularity and the far removed Hawking radiation outside. This means that once the BH has radiated away e^{2S} amounts of information the remaining BH has no corollary information and is thus singular at the horizon = firewall.

    Somehow in order to remove the firewall there must be a subspace of dim ~ e^{2S} in the e^{4S} Hilbert space. I think the uncertainty with the ordering of fields near the horizon means there is an uncertainty concerning the nature of any entanglement. This uncertainty, which is involved with the associator, then means given two entanglements that one thinks should swap, this added uncertainty just means that these two entanglements are different sets of associated operators, similar to sets of commuting operators in QM, and the two entanglements are equivalent by other means than swapping.