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 http://arxiv.org/abs/1303.3935. 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 http://arxiv.org/abs/quant-ph/0101012 .

Hardy’s argument is instrumentalist, but at core this result stems from dimensional analysis for Lie groups as Jochen Rau showed in http://arxiv.org/pdf/0706.2274v1.pdf.

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|

with

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

and

            |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 http://arxiv.org/abs/quant-ph/0104088.

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