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- Xref: sparky sci.physics:21194 alt.sci.physics.new-theories:2550
- Newsgroups: sci.physics,alt.sci.physics.new-theories
- Path: sparky!uunet!well!sarfatti
- From: sarfatti@well.sf.ca.us (Jack Sarfatti)
- Subject: Feynman 35 Vacuum polarization and mental processes.
- Message-ID: <BzAHxv.9ny@well.sf.ca.us>
- Sender: news@well.sf.ca.us
- Organization: Whole Earth 'Lectronic Link
- Date: Tue, 15 Dec 1992 07:25:07 GMT
- Lines: 183
-
-
- Feynman 35 "Theory of Positrons 1" From the future revisted!
-
- *All of this may be important in the physics of our consciousness because
- of virtual electron-positron "Lamb shift" frequency shifts of electron
- transitions in atoms and molecules. These shifts are in the megacycle
- region in hydrogen and are probably smaller in bigger atoms and molecules.
- They could affect neurotransmitter release and capture in synaptic clefts!
-
- 1. Let's take quick (not quite superficial) look at Feynman's classic
- papers on QED. This is mainly meant for non-experts, though pundits may
- learn a thing or two.
-
- "The main principle is to deal directly with the solutions to the
- Hamiltonian differential equations rather than with these equations
- themselves ...The same equation, Dirac's, which describes the deflection of
- the world line of an electron in a field, can also describe the deflection
- ... when it is large enough to reverse the time-sense of the world line,
- and thereby correspond to pair annihilation .... the direction of the world
- line is replaced by the direction of propagation of the waves.
-
- This view is quite different from that of the Hamiltonian method which
- considers the future as developing continuously from out of the past. Here
- we imagine the entire space-time history laid out, and that we just become
- aware of increasing portions of it successively. ... The temporal order of
- events during the scattering, which is analyzed in such detail by the
- Hamiltonian differential equation is irrelevant. ....
-
- ... in non-relativistic quantum mechanics the amplitude for a given process
- can be considered as a sum of an amplitude for each spacetime path
- available .... remove, in the relativistic case, the (non-relativistic)
- restriction that the paths must proceed always in one direction in time ...
-
- 2. Green's function (non-relativistic semi-classical limit in which
- electromagnetic field is classical but the electron is quantum mechanical)
-
- The Schrodinger equation
-
- idpsi/dt = Hpsi (1)
-
- describes change in the wave function psi in an infinitesimal time dt as
- due to the operation of an operator e^-iHdt .... if psi(x1,t1) is the wave
- function at x1 at time t1, what is the wave function t time t2 > t1?
-
- psi(x2,t2) = S[K(x2,t2;x1,t1)psi(x1,t1)d^3x1] (2)
-
- "S" = Integral or Sum depending on context, K is a Green's function
-
- ... if H (i.e., the Hamiltonian) is a constant operator having eigenvalues
- En, eigenfunctions phin, so that psi(x,t) can be expanded as
- S(n)[Cnphin(x)], then psi(x,t2) = S(n)[e^-iEn(t2 - t1)Cnphin(x1). Since
- Cn = S[phin(x1)*psi(x1,t1)d^3x1], one finds (where we write 1 for x1,t1 and
- 2 for x2,t2)
-
- K(2,1) = S(n)[phin(x2)phin(x1)*e^-iEn(t2 - t1)] (3)
-
- fot t2 > t1. We shall find it convenient for t2 < t1 to define K(2,1) = 0
- (Eq. (2) is then not valid for t2 < t1) .... K can be defined by that
- solution of
-
- (id/dt2 - H2)K(2,1) = iDirac delta (2,1) (4)
-
- which is zero for t2 < t1.... H2 means that the operator acts on variables
- 2 of K(2,1). When H is not constant ,,, K is less easy to evaluate than
- (3).
-
- Feynman then gives a physical picture for the Green's function:
-
- We can call K(2,1) the total amplitude for arrival at x2,t2 starting from
- x1,t1. (It results from adding an amplitude e^iS, for each space time path
- between these points, where S is the action along the path.)
-
- I add that K(1,2) is the time reverse of K(2,1).
-
- *Note S = 2pi Classical Action/Planck's constant
-
- Example 1 for non-relativistic free particle phin = e^ipx, En = p^2/2m
-
- Ko(2,1) = S{e^-i[p(x1 - x2) - p^2(t2 - t1)/2m]d^3p (2pi)^-3}
-
- = (2pi im^-1(t2 - t1))^-1/2 e^im(x2-x1)^2(t2 - t1)^-1/2
-
- The transition amplitude for finding a particle in state chi(t2) at time
- t2, if at t1 it was in state psi(t1), is
-
- S[chi*(2)K(2,1)psi(1)d^3x1 d^3x2] (5)
-
- I add note that the integral is over both space-like slices, so that the
- retarded (forward in time) transition amplitude is not a local function of
- space. The advanced transition amplitude backward in time (t2 > t1) is
-
- S[psi*(1)K(1,2)chi(2)d^3x1 d^3x2] (5')
-
- this is not in the original Feynman paper -but I add it because of John
- Cramer's "transactional interpretation" from the old Wheeler-Feynman
- classical action-at-a-distance electrodynamics.
-
- Feynman then develops perturbation theory as multiple scattering of a
- particle in a weak classical potential U(x,t) that differs from zero only
- for t between t1 and t2
-
- K(2,1) = K0(2,1) + K1(2,1) + K2(2,1) + K3(2,1) + ... (6)
-
- and he derives
-
- K1(2,1) = -iS[K0(2,3)U(3)K0(3,1)d3] (9)
-
- note this is an integral over all space and time (i.e., d3 = dx3dt3)
- although in this non-relativistic limit it is only in the slab between
- space-like slices from t1 to t2. Similarly,
-
- K2(2,1) = (-i)^2S^2[K0(2,4)U(4)K0(4,3)U(3)K0(3,1)d3d4] (10)
-
- where S^2 means double integral etc.
-
- 3. Feynman analyses the Dirac equation. K is now a 4x4 matrix operating on
- a 4-component wave function. phin* is replaced by the "adjoint" (i.e., it
- is multiplied by beta matrix from right). However, using the same boundary
- conditions as in the non-relativistic case is the "one electron" theory in
- which the electron scattering off the classical potential U can only
- scatter forward in time with both positive and negative energies. Feynman
- defines energy as "rate of change of phase". "On the other hand, according
- to the positron theory negative energy states are not available to the
- electron after the scattering But there are other solutions .... We
- shall choose the solution defining K+ so that K+(2,1) for t2 > t1 is the
- sum of (3) over positive energy states only. Feynman then proves that since
- K0 - K+ is the sum of (3) over all negative energy states for all times,
- and since K0(2,1) = 0 for t2 < t1, therefore K+(2,1) for t2 < t1 is the
- negative of the sum (3) over negative energy states. Feynman interprets
- the negative sign in terms of the Pauli exclusion principle. "The fact
- that the entire sum is taken as negative in computing K+(4,3) (e.g., see eq
- (10)) is reflected in the fact that in hole theory the amplitude has its
- sign reversed in accordance with the Pauli principle and the fact that the
- electron arriving at 2 has been exchanged with one in the sea.
-
- / 2
- /
- /
- 3 / here is a Feynman diagram for eq.(10)
- /\ /
- / \ /
- / \/ 4
- / virtual pair if t4 < t3
- 1/
-
-
- To this, and to higher orders, all processes involving virtual pairs are
- correctly described in this way.
-
- "The scatterings may, however, be toward both future and past times, an
- electron propagating backwards in time being recognized as a positron.
- This therefore suggests that negative energy components created by
- scattering in a potential be considered as waves propagating from the
- scattering point toward the past, and that such waves represent the
- propagation of a positron annihilating the electron in the potential. ...
- With this interpretation real pair production is also described correctly..
- ... All these amplitudes are relative to the amplitude that a vacuum will
- remain a vacuum, which is taken as unity."
-
- Feynman then discusses the relativistic generalization of eq (2) (i.e., his
- eq. (19)) for the wave function which involves "the closed 3-dimensional
- surfacr of a region of space time containing point 2 the wave function
- psi(2) ... is determined at any point inside a four-dimensional region if
- its values on the surface of that region are specified... That is, the
- amplitude for finding a charge at 2 is determined both by the amplitude for
- finding an electron previous to the measurement and by the amplitude for
- finding a positron after the measurement. This might be interpreted as
- meaning that even in a problem involving but one charge the amplitude for
- finding the charge at 2 is not determined when the only thing known is the
- amplitude for finding an electron (or a positron) at an earlier time.
- There may have been no electron present initially but a pair was created in
- the measurement (or also by other external fields). The amplitude for this
- contingency is specified by the amplitude for finding a positron in the
- future."
-
- *All of this may be important in the physics of our consciousness because
- of virtual electron-positron "Lamb shift" frequency shifts of electron
- transitions in atoms and molecules. These shifts are in the megacycle
- region in hydrogen and are probably smaller in bigger atoms and molecules.
- They could affect neurotransmitter release and capture in synaptic clefts!
-
- to be continued
-
-