Feynman's Cal Tech Course on Path Integrals
recorded by Hibbs.
(McGraw-Hill, 1965, Quantum Mechanics and Path Integrals by Feynman and Hibbs).
Version 0.3
Dec 18, 1996
Part 1
ABSTRACT
This first part is notable in that in 1.6 (on the scattering of neutrons from crystals) we find Feynman's proof of the objective existence of quantum possibilities as a different level of reality from historical actualities. This is an experimental justification for Stapp's ontological theory of inner "felt" quantum consciousness in the outer classical brain based upon the Heisenberg-James meta-theory of the meaning of quantum mechanics. It does not, however, refute Bohm's pilot-wave/hidden-variable meta-theory.
0.1. The infinite self-energy of the classical point electron led Feynman to a least-action principle using half advanced plus half retarded electromagnetic potentials. The source electron sends signals both to the future and the past along its light cone.
0.2. S is Hamilton's principal function defined as the indefinite time integral of the Lagrangian. This is also called "the action". Based on earlier work of Dirac on the analogy of e^iS to the transformation of the quantum wave from one moment to the next, Feynman showed it was more than an analogy. It was an equality.
0.3 It is necessary to use path integrals to describe the time evolution of quantum patterns of information. These integrals over all paths are infinitely multiple integrals over all space variables at each moment of time. The action is a functional of the spacetime path. So these Feynman amplitudes are functional integrals over all possible paths connecting a preparation to a detection.
0.4 The Lorentz invariance of quantum electrodynamics, required by special relativity, is easier to see using path integrals than using traditional the traditional Hamiltonian written in terms of second-quantized creation and destruction operators. This is because the action is a Lorentz frame invariant. Yet, quantum mechanics contains an essential nonlocality which contradicts the spirit of special relativity.
0.5 The path integral is useful to keep track of the infinities of renormalization of the mass, charge and the wavefunction in quantum electrodynamics in a way that is both Lorentz and gauge invariant. Schwinger has an alternate way using canonical transformations and the action principle which is not as visualizable as Feynman's.
1.1 Feynman's informal language is more like Bohr's than Bohm's. For example, in discussing the central mystery of quantum mechanics, i.e., the double slit experiment, we find "We must conclude that when both holes are open, it is not true that the particle goes through one hole or the other." p.6 In fact, in Bohm's theory the hidden variable particle electron always goes through a single slit, but its attached spread out wavy quantum pilot-wave always goes through both slits. The pilot-wave exerts a force on the particle which gives it a wavelike behavior in the region of overlap of the waves from each slit when no other interaction can distinguish which slit the particle actually passed through.
1.2 Wavelike behavior means add the amplitudes of the indistinguishable alternatives before computing the squared modulus. Particlelike behavior means squaring the amplitudes of the now distinguishable alternatives before adding. The squared modulus of the total amplitude has an interpretation in terms of information flowing forward and backward in time. The complex conjugate amplitude is advanced running backward in time from the future detection to the past preparation. Wavelike interference corresponds to information taking different paths forward and backward in time for the same fixed preparations and detections.
1.3 Let a = distance between the two slits. Let L = distance from slits to screen. Let d = distance between successive fringe maxima (i.e., constructive interference) on the screen. Let k = wave number (i.e., inverse wavelength) of the light or matter wave from the slits to the screen. Then, asuming L is large compared to a, so that the two paths, from each slit to the next maximum from the main peak at the center of the screen, are parallel to first order, with a path difference of one wavelength we have by similar approximately right triangles (to first order):
sin@ = 1/ka
d = Lsin@ = L/ka
But the particle momentum associated with the quantum (pilot) wave is the DeBroglie equation p=hk. Therefore, h/d = hka/L, but the component of momentum parallel to the screen is dp = pa/L, therefore, h/d = dp which is the Heisenberg uncertainty principle since d is a measure of the uncertainty in where the photon will land on the screen.
1.4 This is very important on the issue of the role of consciousness in quantum measurement. Feynman refutes Wigner here! "The concept of interfering alternatives is fundamental to all of quantum mechanics. ... suppose that information about the alternatives is available (or could be made available without altering the result), but this information is not used. Nevertheless, in this case a sum of probabilities (in the ordinary sense) must be carried out over exclusive alternatives. These exclusive alternatives are those which could have been separately identified by the information." p.14 Feynman's version of quantum theory is ontological and objective about what is really out there independent of our conscious awareness or knowledge. In this way it differs markedly from Bohr's and Heisenberg's epistemological Copenhagen interpretation of the meaning of quantum mechanics. Feynman certainly does not agree with Wigner's idea that consciousness is required to collapse the wave function from coherent interfering quantum alternatives (i.e., Heisenberg "potentia") to incoherent non-interfering exclusive classical alternatives (i.e.,Heisenberg "actua"). Although, Feynman does not need consciousness to understand quantum measurement, his theory here does not preclude the possibility that quantum mechanics is necessary to explain consciousness. That is, quantum mechanics may be more fundamental than consciousness. Consciousness can be an emergent strictly non-classical quantum-type of phenomenon without contradicting Feynman's meta-theory of the meaning of quantum mechanics. Note there have been recent experiments by Mandel at the University of Rochester which actually demonstrate Feynman's above assertion that one need not actually make a measurement, but that the mere possibility that such a measurement could have been made without changing the observed system, is sufficient to objectively destroy interference between alternatives. Another example of this is given below on neutron scattering from crystals. Note the alternatives always correspond to mutually exclusive classical descriptions of the history of the system.
1.5 Example 1, scattering of two distinguishable nuclei (e.g., an He3 collides with an He4) at right angles in their center of mass frame of reference. One nucleus moves to the right from preparation region A. The other nucleus moves to the left from preparation region B. They collide in the center. One nucleus goes up to detetector 1, the other nucleus goes down to detector 2. There are two alternatives. The Feynman amplitude for the first alternative is the complex number (A1,B2) which has the nucleus from A go to 1, and the nucleus from B goes to 2. Now permute 1 and 2 to get the amplitude for the second alternative (A2,B1) where the nucleus from A goes to 2 and the nucleus from B goes to 1. These are two different classical histories of the pair of nuclei. Classically either one must happen or both. Quantum mechanically both can happen simultaneously because of superposition. This is the quantum weirdness of Schrodinger's Cat where the cat is apparently both alive and dead! All chemical bonds are examples of Schrodinger's Cat. That is, all chemical bonds in the molecules that we are made out of depend on quantum weirdness in an essential way!
Case 1. The nuclei are really different in some way. Then, whether or not the observer has actual knowledge of which path each nucleus takes, the total probability for the two detectors at 1 and 2 to "click" is proportional to the sum of the squared moduli of the complex-valued Feynman amplitudes. That is,
Probability that 1 and 2 detectors both click in a coincidence is proportional to
|(A1,B2)|^2 + |(A2,B1)|^2
Note, for example, that
|(A1,B2)|^2 = (A1,B2)* (A1,B2)
where (A1,B2)* is the complex conjugate of (A1,B2). One can visualize the meaning of the Feynman amplitude (A1,B2) as a retarded process forward in time where one nucleus starts from A in the past and ends at 1 in the future. Similarly, the other nucleus starts at B in the past and ends at 2 in the future. In contrast, the complex-conjugate amplitude (A1,B2)* , that Fred Alan Wolf calls the "star wave" is the advanced process back in time from the future to the past that you get by letting the movie run backwards. That is, the nucleus from 1 in the future moves back in time to region A in the past. Similarly, the nucleus from 2 in the future moves back in time to B in the past. The usual retarded wave is modulated by an advanced wave to create a probability for something to happen. This modulation of a retarded wave by an advanced wave is also called the Born probability density rule. However, for case 1 here, the information runs forward and backward in time over the same alternatives. Each alternative in case 1 is an island unto itself. So, let's go to the more interesting case 2.
Case 2, Two identical Helium 4 nuclei of spin 0. This nucleus has two protons of opposite spins and two neutrons of opposite spins in its lowest energy ground state. Pauli showed that in special relativity, if there is no faster-than-light signalling, and if there is no exotic matter destabilizing the quantum vacuum with negative energy moving forward in time, then all systems with integer spin (in units of Planck's constant/2pi = hbar) obey Bose-Einstein quantum statistics. That is, they tend to attract each other into the same single-particle quantum state possibly forming a Bose-Einstein condensate or superfluid under certain conditions. They also can form coherent and squeezed states. The coherent state is how quantum physics gets what we usually mean by classical waves of sharp amplitude and phase. Squeezed states are very useful for detecting very small signals like gravity waves from pulsars using the giant NASA interferometer out in space.
The Bose-Einstein probability for a coincidence double-click from each detector is now one of constructive interference where
|(A1,B2) + (A2,B1)|^2 = |(A1,B2)|^2 + |(A2,B1)|^2 + (A1,B2)*(A2,B1) + (A2,B1)*(A1,B2)
Note that the new wavelike interference terms correspond to information running forward and backward in time along different alternative histories of the pair of nuclei. In fact, the wavelike behavior of particles generally correspond to a phase connection between the interfering alternatives which link together for the channeling of quantum influences. The new wavelike interference terms are cross-talk or cross-modulations in which an advanced amplitude from one alternative modulates the retarded amplitude of a different alternative.
Case 3 Two identical He3 nuclei of spin 1Ú2. The He3 isotope of helium has only one unpaired neutron in its ground state. Pauli showed that these spin 1Ú2 particles obey Fermi-Dirac quantum statistics of destructive interference. That is, the probability of a coincidence is now proportional to
|(A1,B2) + (A2,B1)|^2 = |(A1,B2)|^2 + |(A2,B1)|^2 - (A1,B2)*(A2,B1) - (A2,B1)*(A1,B2)
where there is a 180 degree phase shift in the pair amplitude interference terms.
Let's do a new experiment. Suppose that we move detector 2 over to the position of detector 1. Let (A1,B2->1) = (A2->1,B1) = (A1,B1), then for the classical case 1 of distinguishable nuclei, the coincident click probability is proportional to 2|(A1,B1)|^2. For case 2 of two identical bosons, the coincidence probability is proportional to 4|(A1,B1)|^2 which is twice the classical result! Even more remarkably, for case 3 of two identical fermions the coincidence probability is exactly equal to zero. This is the Pauli exclusion principle which is always a consequence of Fermi-Dirac statistics when you try to put more than one idnetical fermion into the same single-particle quantum state. The stability and diversity of the matter we are made out of depends critically on this nonlocal quantum principle.
"This rule of the 180 degree phase shift for alternatives involving exchange in identity of electrons is very odd, and its ultimate reason in nature is still only imperfectly understood." p. 16
Feynman's proof of the objective existence of quantum possibilities as a different level of reality from historical actualities. This is an experimental justification for Stapp's ontological theory of quantum consciousness based upon the Heisenberg-James metatheory of the meaning of quantum mechanics.
1.6 Example 2. Scattering of neutrons by a crystal.
"When neutrons of wavelength somewhat shorter than the atomic spacing are scattered from the atoms in a crystal, we get very strong interference effects. The neutrons emerge only in certain directions determined by the Bragg law of reflection, just as for X-rays. The interfering alternatives ... are .. that it is this, or that, atoms which does the scattering of a particular neutron. (The amplitude to scatter neutrons from any atom is so small that we need not consider althernatives in which a neutron is scattered by more than one atom.) The waves of amplitude describing the motion of a neutron which start from these atoms interfere constructively only in certain definite directions. ... Neutrons, like electrons, carry a spin, which can be analysed into two states ... Suppose the scattering material is composed of an atomic species which has a similar spin property, such as carbon-13. In this case an experiment will reveal two apparently different types of scattering ... besides the (coherent no spin-flip) scattering in discrete directions .... there is a diffused (incoherent spin-flip) scattering in all directions ... in order that a neutron flip its spin ... conservation of angular momentum requires that the spin of te scattering nucleus be changed ... The concept of searching through all the nuclei in the crystal to find which one has changed is ... a needle-in-the-haystack type of activity, but nature is not concerned with the practical difficulties of experimentation. The important fact is that in principle it is possible without producing any disturbance of the scattered neutron to determine (in this latter case where the spin states change) which crystal nucleus actually did the scattering. The existence of this possibility means that even if we do not actually carry out this determination, we are, nevertheless dealing with the exclusive (and the non-interfering) alternatives." p.18
1.7 There is still the mystery of how, even in the quasi-classical case of incoherent spin-flip scattering, one classical history actualizes from the entropic mixture of them in any single historical event.
1.8 Summary of probability concepts in quantum mechanics.
Ò... there is a quantity called a probability amplitude associated with every method whereby an event in nature can take place... we can associate an amplitude with the overall event by adding together the amplitudes of each alternative methos ... Next, we interpret the absolute square of the overall amplitude as the probability that the event will happen.Ó p. 19
The absolute square can be reinterpreted as a retarded amplitude propagating forward in time modulated by an advanced amplitude propagating backward in time between the fixed preparation and detection of the overall event. Wavelike interference of localized particles correspond to cross-modulations where the path of the advanced amplitude is different from the path of the retarded amplitude. That is, coherent phase differences between different alternatives link them together as closed stringy loops in time that are boundaries of spacetime sheets of finite area. These stringy loops are not confined to the classical light cone or inside it. In general they correspond to quantum fluctuations from virtual processes. Only the classical limit regions of constructive interference are required to obey the rules special relativity. Furthermore, any extension of quantum mechanics to a new post-quantum mechanics can expect a modification in the rule that the absolute square of the overall amplitude is the probability that an event will actualize. The idea of actualization presupposes the making of an irreversible record that cannot be quantum erased. The arrow of time of the classical second law of thermodynamics enters the picture here. On the other hand, unlike WignerÕs meta-theory of the meaning of quantum mechanics that consciousness of the observer-participator collapses the coherent amplitude into a single actual alternative, Feynman says ÒNoÓ. There is no observer-created subjective physical reality at the quantum level in the case that the observer is trying to get knowledge of a system that is not itself. That is, Feynman does not at all consider the case of Òself-measurementÓ where it can be said that we are responsible for the creation of our own inner psychic reality of felt-experience. What Wigner is talking about, in contrast, would be a kind of psychokinesis between a human mind and a material object that is not the body attached to the mind. The Bohm quantum force is certainly an explanation of how a quantum mind changes the material configuration of its classical brain in a situation of Òself-measurementÓ. Thus, Feynman and Hibbs wrote on the objectivity of external quantum reality that:
ÒIf we interrupt the course of the event before its conclusion with an observation on the state of the particles involved in the event, we disturb the construction of the overall amplitude ... the amplitudes associated with the excluded states can no longer be added in as alternatives in computing the overall amplitude. ... Further, it does not matter if we actually observe and record the outcome of the measurement or not, so long as the measurement equipment is working. Obviously, we could observe the outcome anytime we wished. The operation of the measuring equipment is sufficient to disturb the system and its probability amplitude.Ó p. 19
The key phrase is "Further, it does not matter if we actually observe and record the outcome of the measurement or not, so long as the measurement equipment is working". Feynman does not treat Òinteraction-freeÓ measurements (e.g. RenningerÕs thought experiment) and self-measurement by a sentient measuring apparatus introspecting into its own state in a strange Godel loop. Also note that it is clear that the Heisenberg uncertainty principle was only meanÕt to apply to measurements that are not self-measurements. In fact, David Albert has shown that one can beat the Heisenberg uncertainty principle for special pairs of incompatible observables in a Òself-measurmentÓ which also involves Òphotographs of other worldsÓ violating the dogma of EverettÕs original meta-theory of Òmany worldsÓ for the meaning of quantum mechanics. Everett mistakenly assumed that conscious observers could never be aware of their parallel selves in the Òuniverses next doorÓ.
1.9 Introducing the path integral - structure of the quantum amplitude.
First consider the double slit experiment from a point source S to a pointlike particle detector D fixed at some position on the screen. Let the double-slit plate be called Xa. There are two amplitudes. (S, Xa1,D) and (S,Xa2,D). Now suppose, we drill more slits in the plate Xa. LetÕs drill n of them. We then have the set of na amplitudes {(S,Xaj,D): j = 1, 2, ....na}. The total amplitude to start at the point source S and end up at the detection point D is the sum of all of these amplitudes. Now imagine na getting very large so that the sum approaches the single integral of dXa(S,Xb,D). Next, put in a second plate Xb. Do the same thing by drilling nb slits in the second plate. We now have a set of na.nb amplitudes whose sum approaches the double-integral of dXadXb(S,Xa,Xb,D). Keep increasing the number of plates between S and D. In the idealized continuum limit we get an infinitely multiple integral over all possible paths in space between S and D. We have to include the possibility of paths looping behind S and beyond D. This is FeynmanÕs physical idea of the path integral for all paths that are not discriminated among by some objective interaction independent of our knowledge.
1.10 The amplitudes evolve deterministically from the fixed Schrodinger equation. The probabilities which are the squared moduli express the indeterminism of single quantum events. These are two very different processes. Complete determinism and complete indeterminism co-exist in quantum mechanics as two different objective levels of quantum reality in the standard Copenhagen-type interpretations or meta-theories of the meaning of quantum mechanics. ÒIt is very remarkable that this interpretation does not lead to any inconsistencies .. one never quite loses the feeling that there is something peculiar about the subject.Ó p. 22
1.11 Sarfatti not Feynman: Sentience, subconscious, conscious, and superconscious, cannot happen in quantum mechanics because it is not compatible with either determinism or indeterminism. Sentience, or the physical Òelan vitalÓ, exists on the edge between determinism and indeterminism. It requires a new kind of post-quantum mechanics for living organizations of matter and radiation that limits to quantum mechanics when a certain vital parameter called Òback-actionÓ goes to zero.
1.12 Feynman mentions still unsolved fundamental ÒphilosophicalÓ not ÒphysicsÓ problems. They are:
I. Show that the squared modulus probability axiom is Òthe only consistent interpretation ...Why can we only predict the probability that a given experiment will lead to a definite result? From what does the uncertainty arise? Almost without a doubt it arises from the need to amplify the effects of single atomic events to such a level that they may be readily observed by large systems. The details of this have been analyzed only on the assumption that the [Born axiom] |amplitude|^2 is a probability, and the consistency of this assumption has been shown. It would be an interesting problem to show that no other consistent interpretation can be made.Ó p.22
Bohm, and also Vigier, introduced a sub-quantum level. When the degrees of freedom of that level are in thermal equilibrium he gets the above Born axiom. This suggests that pumped non-equilibrium of the sub-quantum level will correspond to the back-action effects of post-quantum mechanics where the Born axiom, and EberhardÕs theorem (i.e., no superluminal clairvoyance and retroactive precognition via nonlocality is possible in quantum mechanics), break down. Bohm showed in 1952 that even without the sub-quantum level assuming the Born axiom initially along with zero back-action preserves the Born axiom as the system evolves in time. Note, however, that back-action spoils the Born axiom even if it is assumed initially.
II. Ò... there seems to be a lack of symmetry in time of our knowledgeÓ and our felt-experience. ÒOur knowledge of the past is qualitatively different from our knowledge of the future. In what way is only the probability of a future event accessible to us, whereas the certainty of a past event can often be apparently asserted? .... Obviously, we are again involved in the consequences of the large sizes of ourselves and of our measuring equipment. ... What seems to be needed is the statistical mechanics of the amplifying apparatus.Ó