Hameroff-Penrose Quantum Brain

we propose that ... quantum coherence ... and a ... phenomenon of quantum wave function "self-collapse" objective reduction: OR are essential for consciousness, and occur in cytoskeletal microtubules and other structures within each of the brain's neurons ... conformational states of microtubule subunits (tubulins) are coupled to internal quantum events, and cooperatively interact (compute) with other tubulins. We further assume that macroscopic coherent superposition of quantum coupled tubulin conformational states occurs throughout significant brain volumes and provides the global binding essential to consciousness....

Here is a picture of a single tubulin (T) . What follows is my mathematics. It is not in the Hameroff-Penrose paper.

The yellow blobs show two possible spatial configurations (+ and -) of a single electron e that controls the two possible spatial shapes in grey (+) and pink (-) of the single protein dimer molecule. The electron e (+,-) and the protein p (+,-), therefore, form an "entangled" Einstein-Podolsky-Rosen (EPR) pair state |Tj) for the jth tubulin which is

|Tj) = cosxj|ej+)|pj+) + e^iyj sinxj|ej-)|pj-)

where x and y are continuously-variable parameters of the coherent superposition of the electron-protein complex. There are no crossed terms in which a + for an electron appears with a - for a protein etc. Note that the electron does not have a pure coherent wave function of its own. If we integrate out the protein variables, the electron is in an incoherent mixed state. Therefore, the quantum computing switch is the entire tubulin. I have not included the spin of the electron which might be important.

For N tubulins, we have the product state |k):

|k) = Prod(j = 1 to N) |Tj)

in which the particular tubulin configuration k depends on 2N continuously-variable parameters xj and yj. What is the equation corresponding to HP's words "macroscopic coherent superposition of quantum-coupled tubulin conformational states"? It is not simply the multiple integral product state

|product> = Prod(j=1 to N)|Tj)Integral{dxjdyj(Tj|xj,yj)}

because here the different tubulins are statistically independent and there is no nonlocal context-dependent quantum force from the "mental/qualia" (my postulate not HP's) collective pilot wave quantum-gluing two, or more, spatially separated "material" tubulins into a higher-level whole that is greater than the classical sum of its parts.

Let's consider the simplest interesting system of N = 2 tubulins. Define

|Tj+) = |ej+)|pj+) = |j+>

|Tj-) = |ej-)|pj-) = |j->

We now have a more interesting tensor-product entangled state of N= 2 generally quantum-correlated spatially separated tubulins created by the actual gauge forces between the tubulins

|N=2> = |1+>|2+>< 1+2+|N=2> + |1+>|2->< 1+2-|N=2>+ |1->|2+>< 1-2+|N=2>+ |1->|2->< 1-2-|N=2>

Note there are some special choices of the expansion coefficients of measure zero which make this a statistically independent product state of uncorrelated tubulins. However, in general, a single tubulin will not have a pure zero-entropy quantum state of its own. It will locally be in an impure non-zero entropy reduced density matrix - neglecting the quantum gravity friction using only orthodox quantum mechanics. When this pair state is normalized, each coherent superposition is a point on the surface of a unit hypersphere embedded in an 8 real-dimensional Euclidean informational code-space. This is because there are 4 complex expansion coefficients. In general, a state of N tubulins will have a coherent superposition that is a point on a unit hypersphere whose surface of 2^N+1 - 1 dimensions is embedded in a 2^N+1 real-dimensional Euclidean information space. Remember, for orientation, a unit circle in the plane is a unit hypersphere (1-sphere) embedded in a 2 real-dimensional space. An ordinary unit "2-sphere" is embedded in a 3-real dimensional space. Here we only have even-dimensional embedding spaces whose unit-spheres are odd-dimensional.

Quantum Coding Hyperspheres
N tubulins sphere dimension

1 3

2 7

3 15

4 31

Quantum Coding Hyperspheres
N tubulins	sphere dimension
1	3
2	7
3	15
4	31

Note that the quantum-coding hypersphere surface dimension for Penrose's single graviton of spin 2 is (2x2+1)^2 - 1 = 24. This number for an error-correction code seems to come up in Nanopoulos's string theory of quantum gravity objective reduction. This would only apply to a massive graviton because massless gravitons, like all massless particles, only have two states of helicity.