Tunneling-vacuum interaction and time-translation of biomolecules.Quantum Tunneling
It is generally acknowledged that energy is momentarily borrowed from and returned to the spacetime vacuum during quantum particle tunneling. The uncertainty principle allows these undetectable energy exchanges for sufficiently brief time intervals (1). A diagram representing proton tunneling-vacuum interactions is shown in Fig. 1. The total particle traveling time from potential well to potential well is greater than zero (1,2), but traveling time within the barrier is zero (3,4).
The x-axis represents distance and the vertical axis indicates forward-time, +t. Barrier boundaries are represented by a and b. The proton path (dashed line) begins at its initial potential well location [h.sub.1]. Within the barrier (between the two proton positions [h.sub.a] and [h.sub.b]) the proton is represented as interacting with the vacuum via a pair of virtual photons, "[left and right arrow] [v.sub.a]" and [left and right arrow] [v.sub.b]" which are unobservable. Virtual vacuum interactions are simultaneous and characterized as "instantaneous," as is proton motion between [h.sub.a] and [h.sub.b]. The proton's final location is at potential well position [h.sub.2]. Kinetic energy within a tunneling barrier, between positions [h.sub.a] and [h.sub.b], is necessarily negative; otherwise the potential energy in the barrier would exceed the total energy of the system.
The Transactional Interpretation (TI) of the Wheeler-Feynman absorber theory (5-7) is applied to Fig. 1. In this theory, observable photons or particles are created from the mutual collapse of "retarded" State Vector (SV) "Offer" Waves (OW) that propagate forward in time and "advanced" (complex conjugate) SV "Confirmation" Waves (CW) that propagate backward in time. They're exchanged in pairs between an emitter and an absorber. When boundary conditions are satisfied, a "handshake through time" is established. The pair then collapses into a real photon or particle that travels from the emitter to the absorber. The dashed proton path lines in Fig. 1 thus correspond to overlapping TI Offer and Confirmation SV Waves. The virtual photon SV Waves do not collapse. Boundary conditions nonetheless hold for these waves.
In the TI, the spacetime vacuum is described as continuously emitting positive and "negative" Offer and Confirmation Waves of all possible polarizations and energies. These are constituents of the vacuum zero-point fluctuations (ZPF) that interact with the tunneling proton.
Negative energy and Imaginary time
There are two equivalent interpretations of the diagram within the tunneling barrier. One in which "negative" kinetic energy is borrowed from and returned to the vacuum in forward-time, and the other wherein positive kinetic energy is emitted from and returned to the vacuum in reverse-time. The "negative" energy interpretation could be attributed to our forward-time reference frame. Feynman regarded "negative" kinetic energy as positive energy moving in the opposite direction in reverse-time. However, neither real forward-time nor real reverse-time exists within the tunneling barrier. The change occurs because time is a complex-valued scalar having both real (observable) and imaginary (unobservable) perpendicular components.
In Fig. 1, the imaginary-time axis within the barrier (perpendicular to the figure plane) is represented by dots. In the regions outside the barrier, a particle follows a real-time trajectoryatisfying the usual laws of motion. There the complex time vector has an observable non-zero real-time coordinate and an unobservable zero-valued imaginary-time coordinate, respectively (xt, 0it). But within the barrier, the Schrodinger equation has wave solutions which include "negative" energies (as above), imaginary tunneling particle momenta, and Wick rotations (multiplication by i) of complex time vectors (Feynman path integral approach) (8).
[FIGURE 1 OMITTED]
The Wick rotation is a mathematical operation applied to quantum wavefunctions. It connects quantum mechanics to statistical mechanics, and for example, relates the Schrodinger equation to the heat equation. Hawking used the Wick rotation in his work on imaginary time. The Wick operation rotates the real-time vector coordinates (xt, 0it) by [pi]/2 to the perpendicular purely imaginary-time direction, (0t, xit). Observable real-time is zero within the barrier (4). The tunneling proton thus constitutes an instantaneous electric current flow with zero observable energy input. In essence, "negative" kinetic energy within the barrier is synonymous with an imaginary-time "shortcut" through four-dimensional spacetime. The Wick rotation is reversed, (0t, xit) - (xt, 0it), when the particle exits the barrier.
The amount of energy exchanged with the vacuum can be determined by the amount of "negative" energy required to conserve the total energy of the system during tunneling. It can also be found using the Feynman amplitude by first calculating the equivalent momentum exchanged with the vacuum. Proton momentum, p, is complex, with the real momentum coordinates (xp, 0ip) undergoing a change to purely imaginary momentum coordinates within the barrier, (0p, xip). For imaginary momentum, [p.sub.i], proton kinetic energy in forwardtime is "negative", -[E.sub.k] = [([p.sub.i]).sup.2]/2[M.sub.p], where [M.sub.p] is the proton mass. The momentum change mediated by the virtual photon transactions is the (complex) direction difference, [p.sub.i], between the ingoing and outgoing momentum (p [right arrow] pi, pi [right arrow] p) for the tunneling proton at barrier boundaries. The exchanged energy is derived from the momentum by assuming the usual energy-momentum relation for real photons ([E.sub.k] = cp). In that case, the energy is directly proportional to the modulus of the momentum vector (9,10), which for the virtual photons is [absolute value of [p.sub.i]]. "Negative" kinetic energy, -[E.sub.k], is returned to the vacuum in forward-time when (0p, xip) - (xp, 0ip). In reverse-time, positive energy, [E.sub.k], is donated to the proton when (0p, xip) - (xp, 0ip).
Virtual photons conserve energy and momentum during Wick rotation (vacuum energy and momentum contributions after tunneling must equal zero). Consequently, the entire particle-vacuum interaction directly participates in the proton's State Vector loop. The "handshake through time" is prearranged throughout all positions [h.sub.1], the virtual photons, the tunneling region, imaginary-time, and [h.sub.2] before virtual interactions and tunneling commence. Proton energy and momentum at [h.sub.1] and [h.sub.2] are always equal and remain correlated during the vacuum's integral participation.
The SV waves for vacuum virtual photons can accordingly be characterized as follows or barrier positions at [h.sub.a] and [h.sub.b]. Again, energy signatures can be imagined in terms of forward or reverse-time, but neither is expressed within the barrier. For example, each vacuum SV pair initially envisioned in reverse-time--now zero time--constitutes a positive Kinetic Energy (KE) virtual photon transaction:
Vacuum sends positive energy in zero reverse-time to proton at [h.sub.b]:
Vacuum sends positive KE OW to proton ....in zero reverse-time
Vacuum receives positive KE CW from proton.... in zero forward-time
Vacuum receives positive energy in zero reverse-time from proton at [h.sub.a]:
Proton sends positive KE OW.... in zero reverse-time
Proton receives positive KE CW from vacuum.... in zero forward-time
In forward-time, these become:
Vacuum sends "negative" energy in zero forward-time to proton at [h.sub.a]:
Vacuum sends "negative" KE OW to proton.... in zero forward-time
Vacuum receives "negative" KE CW from proton.... in zero reverse-time
Vacuum receives "negative" energy in zero forward-time from proton at [h.sub.b]:
Proton sends "negative" KE OW to vacuum..... in zero forward-time
Proton receives "negative" KE CW from vacuum.... in zero rev erse-time
Thus the proton is viewed as carrying "negative" kinetic energy with it from [h.sub.a] to [h.sub.b] during tunneling in forward-time. Upon collapse of all the proton SV waves (from [h.sub.1] to [h.sub.a], from [h.sub.a] to [h.sub.b], and from [h.sub.b] to [h.sub.2]), the proton is viewed as tunneling with overall positive real-time from h1 to [h.sub.2] while tunneling instantly through the distance from [h.sub.a] to [h.sub.b].
References (3,4) show a range of total tunneling speeds (from [h.sub.1] to [h.sub.2]) for v < c through v > c. Reference (4) explains the Hartman effect in terms of instantaneous tunneling between [h.sub.a] and [h.sub.b]; an increase beyond a given tunneling length is no longer accompanied by an increase in tunneling time (3,4). Despite apparentperluminal tunneling speeds, causality violations (information sent into the past) are precluded during individual tunneling events (4). However, the overall tunneling speed dilates time for a tunneling particle and the dilations can be increased by the Hartman effect.
Time-dilations can play a role in quantum "time-translation" (11-13). This is a hypothetical "weak measurement" quantum process wherein superposed time-dilated systems are changed into their past or future states of evolution (11-13). This procedure was initially proposed only as a thought experiment. Practical means for creating the required time-dilations in a laboratory were not identified at that time. It is now conjectured that, IF time-translation were ever to occur, time-dilations during quantum tunneling might naturally produce such time shifts. This is conjectured to arise very rarely in certain biomolecules.
Time-translation involves a superposition of, in this example, tunneling events that evolve at different rates. It relies on weak measurements (time-dilations) involving the advanced and retarded State Vector waves described above. A time-dilation that occurs during a State Vector exchange alters the evolution of the tunneling system (11). The superposition of differing evolutions can create interferences equivalent to a single process that evolves at an uncharacteristic rate, T', even to the point that the evolution can be in reverse-time (12). T' in turn depends on a particular choice of pre-and post-selection measurements such that the weak value associated with the time T' is either positive or negative (respectively forward in time or backward in time) (12). Negative kinetic energy is increased during these processes.
Left to chance, probability for a successful post-selection measurement for time-translation low (12). Probability is increased (but is still remote) if the expectation value of the energy of the system can be estimated (this is possible for known molecules). The number of superposed tunneling systems is thereby minimized, dependent on the magnitude of energy dispersion (12) (also discoverable for specific tunneling events). Double-stranded microRNA (miRNA), for instance, can host superposed tunneling base-pair protons. This is a gene-regulating molecule active in cell growth, neural processes, etc., with [less than or equal to] 21 base-pair bonds. Tunneling superposition occurs by virtue of vibronic entanglement that arises in the molecule during tunneling (See Appendix A). It tends to isolate the systemom disruptions by the surrounding environment. As shown in the Appendix, these entanglements are stable at body temperatures so are not easily disturbed.
Initial times ([t.sub.0] entering the barrier) and final times ([t.sub.0] + T exiting the barrier) for all superposed tunneling events appear instantaneous to an external observer. The external quantum variables in references (11,12) (existing a time between two measurements) correspond to the time-dilations within the tunneling barriers, in turn determined by the tunneling distance, d, in each bond. The time period T corresponds to different periods of the proper (dilated) times [T.sub.i] for the individual tunneling systems.
A wave function of a tunneling quantum system, [PSI](q, t), typically has a Fourier transform which decreases rapidly for large frequencies (energy dispersion) (12), a trait inherent in tunneling (4). The superposed sum of multiples of that wavefunction, shifted by small periods of time and increased by resulting amplification coefficients, creates a quantum interference phenomenon (12). This results in a wavefunction, [PSI](q, t) [+ or -] [DELTA]t), where [DELTA]t is the possibly large real-time shift of the superposed system (12).
As a consequence of the superposition, the system can move to a realtime [t.sub.0] + T' which may differ significantly from the time [t.sub.0] + T (12). Time [t.sub.0] + T' can correspond to a factual past state (a state in which the system actually was at some time t < [t.sub.0] (12)), a contrafactual past state (the state in which it would have been with undisturbed evolution from some past time t < [t.sub.0] (12)), a factual future state (a state at t < t0 in which the system actually will be at the future time t > T (13)), or a contrafactual future state (the state in which it would have been after undisturbed evolution at some future time t > T (12)). For example, if [DELTA]t > T the system occupies the state it was in before the time-translation process began (12).
Factual time-translation allows instantaneous change from a real-time evolutionary state of a given system into another real-time evolutionary state of itself, where the two different states are normally separated from each other in real-time. This constitutes a time "jump". It requires zero time between past and future events; a condition satisfied by a time-translation involving imaginary-time. Wick rotations during superposed tunneling events are uniquely suited to this task.
The spacetime separation ([DELTA]s) between two points (as per Wick rotation) is given by
[DELTA][s.sup.2] = [DELTA][x.sup.2] + [DELTA][y.sup.2] + [DELTA][z.sup.2] + [c.sup.2] [DELTA][t.sup.2]c2[DELTA][ti.sup.2], (1)
where + c2[DELTA][t.sup.2] is the real-time difference between temporal events and -c2[DELTA][ti.sup.2] is the imaginary-time difference (where ti is imaginary). For any separation in 4D spacetime there's an imaginary-time difference, -c2[DELTA][ti.sup.2], for which the spatial and real-time distances between historical events in 5D spacetime are zero (with imaginary-time being the actual fifth dimension of spacetime) (14).
For tunneling-based time-translation, this condition is satisfied when
0 = [DELTA][x.sup.2] + [DELTA][y.sup.2] + [DELTA][z.sup.2] + [c.sup.2] [(t[+ or - ][DELTA]t).sup.2][c.sup.2][DELTA][ti.sup.2], (2)
where [c.sup.2][(t [+ or -] [DELTA]t).sup.2] is the resultant real-time change made in the system. Thus imaginary-time, [t.sub.i], increases with the translated real-time shift, t [+ or -] [DELTA]t. Additionally, increases in [DELTA]x and [DELTA]t are co-dependent, where [DELTA]x equals the increased tunneling distance, [DELTA]d. The real-time shift, t [+ or -] [DELTA]t, corresponds to an equivalent longer [DELTA]d. It creates a larger imaginary time, -[c.sup.2][DELTA][t.sub.i.sup.2], and increases negative energy (potential energy in the equivalent barrier is higher). (This concurs with Schrodinger solutions showing a negative energy increase with barrier length, [DELTA]d, while the imaginary eigen-value of the wavevector [kappa] is constant, independent of barrier length (the Hartman effect)). Resultant changes in the tunneling event from [h.sub.a] to [h.sub.b] are coordinated with [h.sub.1] and [h.sub.2] by the TI State Vector "handshake through time", conforming the circumstances at [h.sub.1] and [h.sub.2] to the translated time t [+ or -] [DELTA]t.
The equivalent shift in imaginary-time, (-[c.sup.2][DELTA][t.sub.i.sup.2]), can be any one of all possible imaginary-time coordinates. Imaginary-time is spacelike (14,15), hence the resultant real-times may not coincide with states already on our real-timelines. Nonetheless, if there is adequate isolation from the environment, access such alternate past or future times (alternate timeline events) is allowed via Wick rotations.
This suggests an interesting conjecturalescription of how contrafactual time-translations might arise: Reversed Wick rotations return the systems to real-time coordinates beyond the tunneling barriers, causing the alternate times to appear in real-time as contrafactual states. Such a scenario suggests all possible times (and hence their respective event outcomes) are accessible via imaginary-time, but must be rotated into real-time by (reversed) Wick rotation to be observed.
This behavior is consistent with Hawking's description of imaginary-time as being spacelike and perpendicular to the real-time arrow (14,15). During this quantum process, (i) there is direct interaction with the vacuum, (ii) General Relativity fails (v > c), and (iii) time is significantly altered. These actions overtly suggest imaginary time as a potential (five-dimensional) direct route to unification of General Relativity with quantum mechanics. It now appears that if time is considered complex, this unification may be possible (16) and is in agreement with these descriptions. Complex time provides the basis for a physical interpretation of the correspondence between quantum and classical mechanics in terms of quantum decoherence (17). It opens a definite possibility for the instantaneous transmission of information through the theoretical prediction of massless particles traveling at velocities greater than the speed of light (18). The properties also seem to match the "objective reduction" (OR) process-volved in consciousness as deduced by Penrose (19), wherein a cross-over point occurs between the quantum and classical levels.
It seems plausible that, IF time-translation does occur, it may well be at the molecular level in superposed systems of quantum tunneling bonds. Effects ultimately created by time-translations (e.g. trans-time communication, tissue reanimation, etc. (13)) would conceivably be initiated by altering the tunneling parameters within corresponding superposed molecular systems.
Long-range AC electron conductance in DNA (and double-strand RNA) is attributable to water that sheathes the strands, while short-range DC conductance occurs within limited segments of base pair bonds in the pi-stack in the center of the molecule (20,21).
The mechanism of DC electric conductance in DNA and dsRNA involves thermal vibration and base-pair bonds (22), suggesting electronic/ vibration entanglement in these bonds during proton tunneling. A tunneling proton in a base-pair bond creates strong entanglement according to a general estimation formula traditionally applied to superconductors (23):
Vx /Vz [approximately equal to] ([e.sup.-[lambda]]/12[lambda]) ([k.sub.F]d) ([w.sup.2]/[r.sub.0.sup.2])], (3)
where Vx and Vz are respectively the longitudinal (acoustic) and transverse scattering potentials, d is the tunneling length, [k.sub.F]d [approximately equal to] [d.sup.21], [r.sub.0] (in [Angstrom]) is the average extension of the tunneling atom wave function (along the tunneling axis) in the minimum, given by (23)
[r.sub.0] = [([nabla]/M[[omega].sub.3]).sup.1/2], (4)
where the bond frequency is [[omega].sub.3], M is the mass of the tunneling atom, and where the approximate tunneling rate [e.sup.-[lambda]] [approximately equal to] exp([-d.sup.2]/[4r.sub.0.sup.2] (23), and the width w [approximately equal to] [3r.sub.0] (23).
A value of 1 for Vx/Vz indicates a state of maximum entanglement wherein bond electrons and tunneling atoms move jointly and, in a superconductor, constitutes the superconducting state. In perovskytetype materials, for example, the condition of Vx/Vz = 1 arises at ~ 70[degrees] K, where the heavy tunneling atoms create entangled states that propagate throughout the lattice and give rise to long-range electron interactions.
Using DNA/dsRNA base-pair parameters in the above expression yields a value equal to or exceeding 1 (correct to order of magnitude) for tunneling at body temperatures, where the base-pair bond frequencies w3 are [less than or equal to] 1.2 x 1013 Hz (~ 400 cm-1)24, M is the mass of the tunneling proton, 1.672 [yen] 10-27 kg, and the tunneling lengths d [greater than or equal to] 1 [Angstrom].
Entangled tunneling distances increase as base-pair bond frequencies decrease. For example, a bond with frequency [[omega].sub.3] = 62.9 [cm.sup.-1] = 1.8857 x [10.sup.12] Hz has complete vibronic entanglement, Vx/Vz [greater than or equal to] 1, for tunneling lengths of d [less than or equal to] 3.7 [Angstrom]. At [[omega].sub.3] = 46.7 [cm.sup.-1] = 1.4 x [10.sup.12] Hz, any d [less than or equal to] 4.4 [Angstrom] is completely entangled:
Clearly the typical base-pair tunneling distances can fall within the Hartman effect range (3).
Many thanks are due to Cathy Lee, Gunter Nimtz, Karoly Vladar, and Pavel Hobza. Much appreciation is expressed for the perceptive suggestions made by the reviewers.
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