It was physicist Henry Stapp who first noted that human free will is likely bound up to quantum mechanical brain states, and the extent to which a superposition of states can occur in association with such states:

As we know, a quantum wave function U is resolved into component states:

U = U(1) + U(2) + U(3) + U(4) +..........U(n)

which make up what we refer to as a "superposition of states". Thus, the vector sum of all the component wave states are integrated into the single wave state U. Stapp, for his part, has pointedly noted that uncertainty principle limitations applied to calcium ion capture near synapses shows they (calcium ions) must be represented by a probability function.[1] More specifically, the dimension of the associated calcium ion wavepacket scales many times larger than the calcium ion itself. This nullifies the use of classical trajectories or classical mechanics to trace the path of the ions. But does it nullify determinism - or more specifically acausal determinism? This is the question we must address.

In the depiction being formulated here, I assume in line with modern neuroscience, that brain dynamics and function is contingent upon the neuron and its connections to synapses. Thus we want networks with the prescriptions given – i.e. quantum units that invoke

A neuron in sub-complex 'A' either fires or not. The 'firing' and 'not firing' can be designated as two different quantum states identified by the numbers 1 and 2. When we combine them together in a vector sum diagram, we obtain the superposition:

U (n ( A] = U (n1 ( A1] + U(n1 ( A2)

where the wave function (left side) applies to the collective of neurons in 'A', and takes into account all the calcium wavepackets that factored into the process. What if one has 1,000 neurons, each to be described by the states 1 and 2? In principle, one can obtain the vector sum as shown in the above equation for all of the neuronal sub-complex A, and combine it with all the other vector sums for the sub-complexes B, C, D and E in an optimized path. The resulting aggregate vector sum represents the superposition of all subsidiary wave states and possibilities in a single probability density function. This allows the (theoretical) computation of the density function, as well as distinct probability amplitudes for the various sub-complexes.

As we’ve seen, application of the Heisenberg Uncertainty Principle to Ca+2 ions (at body temperature) discloses the associated wave packet dimension increases to many times the size of the ion itself.[3] Any classical Newtonian mechanics is therefore inapplicable and irrelevant. Worse, use of such – say for the ions’ trajectory, certainly ensures an erroneous result. Thus we represent the ion uptake superposition as a separate contributor to the aggregate assembly:

U (n (A, B…E) = {U(Ca+2) [ni(A), nj(B)…nm(E)+.............nm(N)]}

wherein all possible states are taken into account.

The total of these taken in concert enables a quantum computer modality to be adopted for a von Neumann-style consciousness. In quantum neural networks it is accepted that the real world brain generation of consciousness is more along the lines of a quantum computer-transducer than a simple collective of switches. As S. Auyang has observed, consciousness is more than a ‘binary relation between a Cartesian subject and object’.[4]

Because a binary relation doesn't cut it, it leaves the door open to an acausal unfolding of brain processes. So, how would quantum acausal determinism enter?

In terms of the example given in Stapp (book already referenced):

[A_n, L (Ca++)] = -i ħ

and the neuronal assembly expectation value (neuron firing sequence initiated at nth A neuron) mutually interferes with the wave packet dimensions (L) of the nth Ca++ ions. Now, let an assembly of neurons (A_n) then be operated on by a set of hidden variables, e.g. φ{A_n} such that we obtain a P-wave packet (pilot wave packet comprised of an ensemble of de Broglie waves or B-waves)

φ{A_n} = d exp [iS/ ħ]

where d is the vector wave density and S a common phase for the P-waves. Then the assembly A_n moves with a group velocity v = df/(d(1/lL where f is the group frequency and L the wavelength.

A_n must also satisfy:

[A_n - (m^2 c^2/ ħ^2) ]A_n = 0

Where the bracketed quantity is none other than the "clock guidance" feature from the de Broglie pilot wave hypothesis, hence the deterministic underpinning of Bohmian QM.

(REM: ħ= h/ 2 pi, where h is Planck’s constant or ~ 6.62 x 10-34 J-s)

One solution of the preceding can be shown to be:

A_n (b) = d exp[2 pi (i) [ft – (x/L) + phi]

where f is frequency, t is time, L is wavelength and x is position in direction of wave propagation. f is the phase angle. (For generality, one can phi assume = 2 pi)

These vector waves would have a superluminal velocity. But to put it in terms more apropos of David Bohm's Stochastic QM concept - we instead say that the waves interconnect with particles at a higher dimensional level.

Assume now the total set of one's thoughts contains waves of frequencies ranging from f' (highest) to f, then the quantum potential V_Q can be expressed:

V_Q = h(f' - f), where h is Planck's constant.

Thus, V_Q has units of energy as the other potential functions in physics, e.g. gravitational and electrostatic. On average, the greater the number of possible states, the greater the difference (f' - f) and the greater the quantum potential. In general,

V_Q= { - ħ^2/ 2m} [grad R]^2 / R

Of course, in a real human brain, we have a "many-particle" field (especially since we're looking at neuronal complexes) so that the quantum potential must be taken over a sum such that: V_Q= { - ħ^2/ 2m} Sigma_ i = [grad R_i]^2 / R

The velocity of an individual B-wave is expressed by:

v(B)= grad S/m

Where m is the mass of the particle associated with the B-wave, and S is a phase function obtained by using: U = R exp( iS/ħ)

Where R, S are real. Thought then occurs with the collapse of the wave function U and the onset of a new phase function S’ as a result, such that the B-waves in an original P-wave packet can become dislodged and arrange as a modulated waveform.

But what of a deterministic thought? Is there a particular condition needed - let's call it a sufficient condition for acausal determinism? I believe there well might be and may enter at the following nexus:

Let the associated de Broglie wavelength of a particular Ca+2 ion be defined:

L = ħ/ mgv

where g is the Lorentz factor and m the mass. The angular frequency (w) can be written:

w = mg c^2/ ħ

and the phase velocity:

v(p) = Lw = c^2/ v

Now, in practice a particle has some finite extension, dx and in a certain limit:

dx *d(1/L) ~1

must be represented as a superposition of waves. In this case also, the interference of observables emerges,via:

[x, p] = -i ħ

However, if the hidden variable φ operates, viz. φ{A_n} = d exp [iS/ h] , it is conceivable that dx may achieve a spread which renders superposition of states no longer applicable, say for a tiny increment or uncertainty in time, dt. In this case, a deterministic acausal quantum potential may act with energy:

dE ~ ħ/ dt

It is this energy that can be available to drive a "deterministic" thought. Clearly also, if superposition of states is what elicits "free will" - then that attribute may conceivably vacate for that interval dt.

The condition on the critical spread to achieve this can be expressed:

dx ~ ħ/ d(mg v)

so if the velocity v approaches a critical threshold such that v -> 0, we can see dx spread out to elicit deterministic thought.

As we know, a quantum wave function U is resolved into component states:

U = U(1) + U(2) + U(3) + U(4) +..........U(n)

which make up what we refer to as a "superposition of states". Thus, the vector sum of all the component wave states are integrated into the single wave state U. Stapp, for his part, has pointedly noted that uncertainty principle limitations applied to calcium ion capture near synapses shows they (calcium ions) must be represented by a probability function.[1] More specifically, the dimension of the associated calcium ion wavepacket scales many times larger than the calcium ion itself. This nullifies the use of classical trajectories or classical mechanics to trace the path of the ions. But does it nullify determinism - or more specifically acausal determinism? This is the question we must address.

In the depiction being formulated here, I assume in line with modern neuroscience, that brain dynamics and function is contingent upon the neuron and its connections to synapses. Thus we want networks with the prescriptions given – i.e. quantum units that invoke

*Pauli spin operators*[2] as effective gates, junctions, along with connecting these to each other through a multitude of neural sub-networks. The diagram shown illustrates the two types of concept to be interwoven: biological networks (left side) and an associated quantum vector superposition in terms of wavefunctions .A neuron in sub-complex 'A' either fires or not. The 'firing' and 'not firing' can be designated as two different quantum states identified by the numbers 1 and 2. When we combine them together in a vector sum diagram, we obtain the superposition:

U (n ( A] = U (n1 ( A1] + U(n1 ( A2)

where the wave function (left side) applies to the collective of neurons in 'A', and takes into account all the calcium wavepackets that factored into the process. What if one has 1,000 neurons, each to be described by the states 1 and 2? In principle, one can obtain the vector sum as shown in the above equation for all of the neuronal sub-complex A, and combine it with all the other vector sums for the sub-complexes B, C, D and E in an optimized path. The resulting aggregate vector sum represents the superposition of all subsidiary wave states and possibilities in a single probability density function. This allows the (theoretical) computation of the density function, as well as distinct probability amplitudes for the various sub-complexes.

As we’ve seen, application of the Heisenberg Uncertainty Principle to Ca+2 ions (at body temperature) discloses the associated wave packet dimension increases to many times the size of the ion itself.[3] Any classical Newtonian mechanics is therefore inapplicable and irrelevant. Worse, use of such – say for the ions’ trajectory, certainly ensures an erroneous result. Thus we represent the ion uptake superposition as a separate contributor to the aggregate assembly:

U (n (A, B…E) = {U(Ca+2) [ni(A), nj(B)…nm(E)+.............nm(N)]}

wherein all possible states are taken into account.

The total of these taken in concert enables a quantum computer modality to be adopted for a von Neumann-style consciousness. In quantum neural networks it is accepted that the real world brain generation of consciousness is more along the lines of a quantum computer-transducer than a simple collective of switches. As S. Auyang has observed, consciousness is more than a ‘binary relation between a Cartesian subject and object’.[4]

Because a binary relation doesn't cut it, it leaves the door open to an acausal unfolding of brain processes. So, how would quantum acausal determinism enter?

In terms of the example given in Stapp (book already referenced):

[A_n, L (Ca++)] = -i ħ

and the neuronal assembly expectation value (neuron firing sequence initiated at nth A neuron) mutually interferes with the wave packet dimensions (L) of the nth Ca++ ions. Now, let an assembly of neurons (A_n) then be operated on by a set of hidden variables, e.g. φ{A_n} such that we obtain a P-wave packet (pilot wave packet comprised of an ensemble of de Broglie waves or B-waves)

φ{A_n} = d exp [iS/ ħ]

where d is the vector wave density and S a common phase for the P-waves. Then the assembly A_n moves with a group velocity v = df/(d(1/lL where f is the group frequency and L the wavelength.

A_n must also satisfy:

[A_n - (m^2 c^2/ ħ^2) ]A_n = 0

Where the bracketed quantity is none other than the "clock guidance" feature from the de Broglie pilot wave hypothesis, hence the deterministic underpinning of Bohmian QM.

(REM: ħ= h/ 2 pi, where h is Planck’s constant or ~ 6.62 x 10-34 J-s)

One solution of the preceding can be shown to be:

A_n (b) = d exp[2 pi (i) [ft – (x/L) + phi]

where f is frequency, t is time, L is wavelength and x is position in direction of wave propagation. f is the phase angle. (For generality, one can phi assume = 2 pi)

These vector waves would have a superluminal velocity. But to put it in terms more apropos of David Bohm's Stochastic QM concept - we instead say that the waves interconnect with particles at a higher dimensional level.

Assume now the total set of one's thoughts contains waves of frequencies ranging from f' (highest) to f, then the quantum potential V_Q can be expressed:

V_Q = h(f' - f), where h is Planck's constant.

Thus, V_Q has units of energy as the other potential functions in physics, e.g. gravitational and electrostatic. On average, the greater the number of possible states, the greater the difference (f' - f) and the greater the quantum potential. In general,

V_Q= { - ħ^2/ 2m} [grad R]^2 / R

Of course, in a real human brain, we have a "many-particle" field (especially since we're looking at neuronal complexes) so that the quantum potential must be taken over a sum such that: V_Q= { - ħ^2/ 2m} Sigma_ i = [grad R_i]^2 / R

The velocity of an individual B-wave is expressed by:

v(B)= grad S/m

Where m is the mass of the particle associated with the B-wave, and S is a phase function obtained by using: U = R exp( iS/ħ)

Where R, S are real. Thought then occurs with the collapse of the wave function U and the onset of a new phase function S’ as a result, such that the B-waves in an original P-wave packet can become dislodged and arrange as a modulated waveform.

But what of a deterministic thought? Is there a particular condition needed - let's call it a sufficient condition for acausal determinism? I believe there well might be and may enter at the following nexus:

Let the associated de Broglie wavelength of a particular Ca+2 ion be defined:

L = ħ/ mgv

where g is the Lorentz factor and m the mass. The angular frequency (w) can be written:

w = mg c^2/ ħ

and the phase velocity:

v(p) = Lw = c^2/ v

Now, in practice a particle has some finite extension, dx and in a certain limit:

dx *d(1/L) ~1

must be represented as a superposition of waves. In this case also, the interference of observables emerges,via:

[x, p] = -i ħ

However, if the hidden variable φ operates, viz. φ{A_n} = d exp [iS/ h] , it is conceivable that dx may achieve a spread which renders superposition of states no longer applicable, say for a tiny increment or uncertainty in time, dt. In this case, a deterministic acausal quantum potential may act with energy:

dE ~ ħ/ dt

It is this energy that can be available to drive a "deterministic" thought. Clearly also, if superposition of states is what elicits "free will" - then that attribute may conceivably vacate for that interval dt.

The condition on the critical spread to achieve this can be expressed:

dx ~ ħ/ d(mg v)

so if the velocity v approaches a critical threshold such that v -> 0, we can see dx spread out to elicit deterministic thought.

[1] Stapp, Henry, P.: 1993,

**, Springer-Verlag, p. 42.**

*Mind, Matter and Quantum Mechanics*[2] Pauli spin matrices are: σx = (0,1¦1,0); σy = (0,-i¦i, 0); σz = (1, 0¦0, -1). (Note each left pair is a matrix 'top' and each right pair a matrix 'bottom' - since they are usually written in a rectangular array form.

[3] Stapp, H.: ibid.

[4] Auyang, S.: 1995,

**?, Oxford Univ. Press, p. 112.**

*How is Quantum Field Theory Possible*
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