A SUPERCONDUCTING-LIKE HAMILTONIAN FOR STATE VECTOR REDUCTION
P. S. Petersen
Center For Higher Education (C.H.E.), San Ramon CA
ABSTRACT: If a non-linear term similar to the first-order Ginsburg-Landau superconducting free energy is added to the Schrödinger equation as a white noise Hamiltonian, it chooses a random phase for each possible quantum state, and gives an explanation for Pearle's (1976) suggestion that this might drive state reduction. However, this formulation does not add an anti-linear term to quantum theory. It indicates 'collapse' of the wave function might be accomplished by a superconducting-like non-linear, non-local phase transition in the perceiving 'plenum' (non-locally interacting brain cells, consciousness, etc.). The order parameter is the particular 'universal' wave function. This model is called the Quantum Phase Ensemble Phase Transition (QPEPT).
It has been understood for some time that quantum theory, as expressed by the Schrödinger equation alone, is incomplete. It does not allow for the selection of a unique Hilbert state in a micro quantum observation. Several popular suggestions have been made for the 'interpretation' of solutions of the equation: The 'minimal' or Copenhagen Interpretation (1), the Hidden Variable Hypothesis (2), and the Many Worlds Hypothesis (3), but none of them makes the selection of the realized state an 'actual process' that someday may be tested by observation (4), although attempts at applying stochastic theories to effect dynamical reduction have been made (5).
Pearle (Version I, ref. 6, 1976) suggested that random phases associated with eigenvectors can be used as initial conditions for a momentary non-linear version of the Schrödinger equation which reduces the state vector with appropriate probability. He did not specify, however, the process which selects these initial phases, except that in a later work (ref. 7, 1985) he indicated that it could be the result of interaction with the environment, as Zeh (8), Zurek (9), and Gell-Mann and Hartle (GM&H) (10) have indicated, or a macroscopic measuring device (we would add: 'of which the observer is a vital part'), as in the theory of Daneri, Loinger, and Prosperi (11).
There is a later version of Pearle's theory (Version II, ref. 12, 1979) in which a rapidly fluctuating antilinear term was added to Schrödinger's equation. We will find that equations similar in form to Pearle (II) may be derived without adding an anti-linear term, though we will first show that equations similar to Pearle (I) result from this formulation. These considerations may be helpful in examining a possible 'physical' process triggering reduction.
Bialnicki-Birula and Mycielski (BB&M, ref.13) suggested that a non-linear term could be added
to the Hamiltonian which gave the following form to the Schrödinger equation:
(ih/2π)(¶/¶t)Y(r, t) = [-((h/2π)2/2m) + U(r, t) + F(|Ψ|2)]Ψ(r, t). (1)
The magnitude of such a term was restricted to be small (14) (15), but might still dominate once the interaction of the quantum system with the measuring device is complete. BB&M chose a separable form for the non linear potential. However, with the clearer understanding of the Bell theorem and its implications, it is perhaps more desirable for a non-linear potential to have a non-separable (non-local) character, as in Pearle (Version III, ref. 16, 1984). Such non-separable terms are found in the free energy density for the Ginsburg-Landau macroscopic theory of phase transitions as applied to superconductivity.
The appearance of a single phase in the superconducting (and superfluid) macroscopic order parameter breaks the gauge symmetry implied by a separable linear Schrödinger equation. We take this to mean that we are impelled to adapt quantum theory to include gauge symmetry breaking, that is, if we take the superconducting order parameter to be a quantum wave function in the literal sense. This paper offers a suggestion in this direction.
Dynamical state reduction should resolve the following problems in quantum physics:
1. The macroscopic body problem. We would like to have classical trajectories as a correspondence limit for macroscopic objects. In the theory of Ghirardi, Rimini, and Weber (GRW) (17), the localization process is proportional to the number of particles involved, avoiding quantum mixtures for large objects. This is true in our theory as well.
2. The macro quantum state problem. The macroscopic 'quantum order' observed in superconducting, superfluid, SQUID, and possibly quasicrystals should be an implication of the theory. This requirement is not usually considered, as the claim that, for example, the superconducting order parameter is a quantum wave function is merely an operational hypothesis in Landau-Ginsburg theory. The theory presented here hypothesizes an ensemble of sets of eigenvector quantum phases undergoing 'phase transitions' suggested by those encountered in superconductors and superfluids, and analogous to 2D ferromagnets with non-local interactions.
3. The trigger problem. It would be gratifying to resolve the mystery of state vector reduction by finding a physical process associated with it. GRW spontaneous localization is an attempt to open the discussion, but does not give us a physical trigger for localization. QPEPT suggests the possibility that complexes in the 'plenum' may be involved in selecting a quantum phases for each possible state, and that they mimic Cooper pairs in a superconducting medium.
4. The special relativity problem. A good reduction theory should be Lorentz invariant, since we perform experiments for the most part, in nearly flat Minkowski space. Pearle's theories (I and II) are, and thus QPEPT qualifies as well, since it dovetails with Pearle. For similar reasons, superluminal signaling is prohibited in the theory.
The QPEPT theory meets all four requirements for a state vector reduction theory, whereas the three prime candidates, GRW , Pearle, and GM&H, meet only two.
In the Ginsburg-Landau (G-L) theory of superconductors (18) (19), a macroscopic theory, the condensed phase is at least identical in form to a 'macroscopically occupied quantum state', . In this paper we call the assumption that this is truly a quantum wave function the 'reality hypothesis'. We will assume that there is a small BB&M term added to every quantum Hamiltonian corresponding to the effect of the observational consciousness or brain function, which behaves in the same way as the G-L first order free energy:
Hb = -b |Ψ|2. (2)
We make use of the analogy between macroscopic and microscopic behavior in the sense Tilley and Tilley (20) have described: that there is "an ensemble with all possible values of (phase)" for the macroscopic order parameter, . (Schrieffer does this for the microscopic case for Cooper Pairs, ref 21). We expand the macroscopic 'phase ensemble' for a single eigenvector, as in Pearle,
Φn = Σk cnkφnk exp(-iEnk2πt/h + iθnk). (3)
Where the θnk's are randomly chosen phases for the eigenstate in an ensemble . If we let
an(t) = xn1/2.eθn(t) = <Φn(t) |Ψ (t)>, (4)
Hamiltonian Hb becomes
-b |Ψ|2 = -b Σp,q xp1/2 xq1/2 Σr,s cpr cqs ei(θpr- θqs) φqs* φpr. (5)
Since phases for separate basis states are uncorrelated, we obtain,
Hb = -b Σp xp Σr,s cpr cqs cos(θpr - θps)φps* φpr. (6)
Note that the minimum of this Hamiltonian is for θpr = θps = θp , that is, it breaks phase symmetry and chooses a phase randomly for each basis state. Also note that for a given basis state (a given p) the Hamiltonian is phase-functionally identical to the superconducting G-L Hamiltonian for Cooper Pairs and the XY Hamiltonian for 2D ferromagnetism with non-local interactions. This means that one of the consequences of this theory is the phase coherence of the superconducting medium, and also implies that we can utilize the roughly appropriate term 'phase magnetization', though the interactions are non-local.
The interaction picture Schrödinger equation is therefore
(ih/2π)dan/dt = - <Φn(t)| b |Ψ|2 | Ψ(t)> = -b Σm xm1/2 Σk eiθkm Σp xp Anmp, (7)
Anmp = Σr,s cpr cps <&Phi:n | φps* φpr | Φm>. (8) Equation (4) now yields,
dxn/dt = (4πb/h) Σm xm1/2 xn1/2 sin(θn - θm) Σp xp Anmp (9)
dθn/dt = -(2&pi:b/h) xn-1/2Σm xm1/2 cos(θn - θm) Σp xp Anmp. (10)
These equations are identical to Pearle's for Version I, except that he has a matrix element Anm, where we have Σp xp Anmp. Note that the sign in equation (9) is correct for a Pearle reduction equation, and that we have obtained it without 'adding' to quantum theory.
In these equations the probability, xj, for one quantum eigenvector is quickly driven to one, while the others are driven to 0, and the probabilities will correspond to those in ordinary quantum theory. Pearle's Version I considers the quantum phases as hidden variables which drive reduction. In a simple experiment with polarized light, Papaliolios (22) has found that hidden variable relaxation must occur in less than about 10-14 sec, so that the constant b must be adjusted appropriately.
A STOCHASTIC THEORY:
The above considerations apply only for non-fluctuating Hamiltonians. More generally, we should consider a stochastic version. Pearle (Version II) considers the ordinary Schrödinger equation with a 'white noise' Hamiltonian,
(ih/2π)dan/dt = Σm Hnm am, (12)
is the interaction picture Schrödinger equation. Such an equation would imply reduction of the state vector if the sign were changed in the corresponding amplitude equation,
dxn/dt = (4πb/h) Σm xm1/2 xn1/2 αnm sin(θn - θm - γnm), (13)
Hnm = αnm exp(iγnm) (14)
Ψ = Σm xm1/2 eiθm Φm. (15)
are white noise "Brownian Motion' functions defined by the following conditions:
1. is Hermitian.
2. < > = 0.
3. <(t')(t)> = δnn' δmm' δ(t' - t). (16)
< > is taken over an ensemble of repeated experiments. We shall show that the matrix element of the G-L Hamiltonian satisfies these conditions.
It is clear that a change in the sign of the Hamiltonian occurs at the critical temperature, as the constant b is proportional to T - Tc. This will change the sign in the Pearle amplitude equation, and thus we obtain the right behavior without using an antilinear function.
Our basic assumption is that each preferred basis state may be expanded in a manner similar to that for the superconducting ground state. We allow, as in Pearle (23), that the interaction of the microscopic system with the apparatus or environment may not only produce random phases but a differing amplitude for each phase. The interaction produces indistinguishable microscopic energy states in the apparatus or environment which are not seen in 'pointer' readings or observations. This situation has been investigated by Daneri, et. al. (24) and Zeh and Zureck (25). Though this is a less restricted expansion than in the superconducting case, it is easily seen that the more restrictive expansion allows the argument as well. It is easier to follow the argument put forth by Pearle, therefore we expand the eigenfunctions as in equation (3). This yields
= <φn,k| H |φm,j> exp[-i2π(En -Em)t/h], (18)
and, as in Pearle, is a white noise function.
Now we expand the Hamiltonian, -b |Ψ|2:
H = -b Σpq xp1/2 xp1/2 Σr,s cpr cqs ei(θpr- θqs) φqs* φpr. exp[-i2π(Eqr -Eps)t/h]. (19)
= Σpq Mpqnm , (20)
Mpqnm = xp1/2 xp1/2 <φn | φp φq |φm>exp[-i2π(En -Em + Ep -Eq)t/h], (21)
and where is a white noise function. The argument is identical to Pearle's.
For a reducing equation, we must show that equation (11) applies. This will be true if
Σpq Mpqnm = Anm , (22)
where is a white noise function.
Two theorems will suffice to prove this:
Theorem I: Let Σp Mp = A . If are white noise functions, will be also.
Proof: (of the three white noise requirements)
1. The Hermitian Property is obvious.
2. <> = 0 implies Σp Mp <> = A<> = 0, yielding <> = 0.
3. < (t') (t)> = δ(t' - t). = Σp Mp /A implying
<(t')(t)> = <Σp Mp (t) Σq Mq (t)>/A2
= Σpq Mp Mq δpq δ(t' - t)/A2 = δ(t' - t), for A2 = Σp Mp2.
Theorem 2: Let =. If and are white noise functions, then will be also.
Proof: Properties 1 and 2 are trivial.
3. <(t') (t')(t) (t)> = <(t')(t)>< (t') (t)>
= δ2(t' - t) = δ(t' - t).
It is thus clear that the first term of Ginzburg-Landau theory (inspired by superconductivity), when appled to the Schrödinger equation, produces a Pearle Version II type stochastic reducing equation for the wave function. This piece added to the Hamiltonian is non-linear and non-local (non-separable).
The free energy representing this ensemble of possible basis state phases is analogous to the free energy for a superconductor (26), and reduces to it when the phase ensemble amplitudes are identical . Note that as a quantity one could call the 'quantum noise' (analogous to temperature) is lowered, a phase transition occurs in the quantum phase ensemble which results in the lining up of all phases for a given state. As the quantum noise 'temperature', T -> 0, each 'site' in the ensemble indicates the same collection of phases. Thus the measurement process, which is tied to reaching a critical 'temperature', Tc, will reflect one unique ray in the Hilbert subspace corresponding to a single measurement or 'world' (in the sense of Everett's many worlds). It is suggested that this noise temperature may be associated with the random firing of neurons or microtubule jostling of electrons in the brain. Discussion of this interesting possibility is reserved for the accompanying article being submitted to the Journal of Mathematical Psychology.
The collapse of the wave function in this interpretation is thus modeled as being initialized in a quantum phase ensemble, a picture suggested by and consistent with the literal interpretation of the order parameter as a wave function in the G-L Theory of Superconductors.
Applications of the QPEPT theory are being investigated. Other possible quantum macro-objects, such as quasicrystals (27), for example, may be aligned by a term analogous to an externally applied magnetic field. We call this field, 'the quantum phase field', and descriptions of its application awaits future papers. Suffice it to say that the quantum phase ordering may appear in quasicrystals as nonlocally-induced bond orientational order. Environmental conditions may be built into this field as well, ala Bohm and Bub (38). It is also possible that semiclassical quantum gravity may be redeemable.
In QPEPT theory it is notable that the link between the pre-measurement Hilbert Structure of possible results, and the post-measurement Hilbert structure representing the observation is the
interaction Hamiltonian between phases in the quantum phase ensemble. This is an energy description
of state vector reduction, and thus is a possible solution to what R. Penrose calls 'the minimizing
problem' (29). As mentioned before, QPEPT shares with GRW theory the concept of reduction and
separability dependent on the number of particles. It remains, however, for such a structure and
ordering to be found in the brain or consciousness.
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28. Ref. 2.
29. Ref. 4.