Wednesday, March 03, 2021

Feynman's path integral & superpositions of histories

What happens in between is "everything", Feynman diagrams are another basis of the "space of histories"

In January, I was asked on Quora What happens in between the initial and final states in quantum mechanics. Well, everything happens and nothing happens or it's meaningless to say what happens. A basic principle of quantum mechanics is that the results of the measurements of observables \(L\), i.e. statements \(L=\lambda_i\) obtained by measurements, are the only facts. Quantum mechanics is a set of rules to predict or derive or explain some facts (or their high probability) from other facts. A theory in classical physics could be given this description but also a "more specific one", a classical model of something, but the more specific one has been impossible after the discovery of quantum mechanics.

Feynman's path integral approach to quantum mechanics is a very explicit way to see "in what sense" everything happens. Assuming some initial and final states (in a position-like basis, to simplify the questions about the bases of initial and final Hilbert spaces), the probability amplitude for a transition (i.e. the complex number that determines the matrix element of a unitary evolution matrix) may be calculated as\[ {\mathcal A}_{{\rm initial}\to {\rm final}} = \int{\mathcal D}x(t)\,\exp(iS/\hbar) \] where the infinite-dimensional integral goes over all trajectories \(x(t)\) connecting the points \(x(t_{\rm initial}) = x_{\rm initial}\) and \(x(t_{\rm final}) = x_{\rm final}\). If you consider generic initial wave functions, you must pre-add two additional integrals over \(x_{\rm initial}\) and \(x_{\rm final}\), the weights \(\psi_{\rm initial}(x_{\rm initial})\) and \(\psi_{\rm final}(x_{\rm final})\), and then the integral becomes the integral over literally all functions \(x(t)\) defined on the interval of time between the initial and final time (be sure that while I admire \(\rm\TeX\), it's faster for me to write these mathematically heavy sentences in words than in mathematical symbols!).

Paul Dirac already pre-invented the path integral by realizing that the unitary evolution matrices over infinitesimal periods of time may be "composed" to a chain and the resulting integrals get combined to an integral over paths. However, he didn't elaborate upon these things. Richard Feynman did. And he used the path integrals to derive the Feynman diagram industry in quantum field theory – which are "the" tool to calculate almost anything in perturbative quantum field theory and beyond (although the path integrals are equally applicable for non-relativistic quantum mechanics of point-like particles, including the hydrogen atom problem which is solvable etc.).

You should notice that the only "process-independent" factor in the path integral's integrand is \(\exp(iS/\hbar)\). Because the action of the trajectory \(S\) is real (this reality condition is linked to the reality of energy or the conservation of the total probability), this exponential is a "complex unity". It always has the same absolute value (well, it is one if written in this way but an extra normalization pre-factor should be added to the path integral). It means that every trajectory, however non-classical or far from an "expected" trajectory, contributes the same to the path integral, up to a phase!

But a deep point of the path integral and quantum mechanics in general is that the relative phases matter. In Feynman's path integral approach to quantum mechanics, the relative phases are the only thing that matters! The whole dynamics (the formulae for the kinetic and potential energy, and therefore the forces etc.) influences the final results through the phase only because the phase is simply \(iS/\hbar\), the action for a history, and it is the only object that knows about the formulae for the kinetic and potential energies and other things!

So how does it happen that the phase guarantees that in a classical limit, the "reasonable" trajectories near the classical path "win"? They win because... they have almost the same phase and that is why the terms from them constructively interfere! Equivalently, the hopeless histories have random phases which is why their complex numbers almost precisely cancel (destructive interference). Indeed, a classical trajectory is one that solves the classical equations of motion and they may be derived from the extremization of the action,\[ \delta S = 0. \] But the statement that the "variation of the action is zero" really means that the nearby trajectories for the classical trajectory have the same \(S\) as the classical trajectory itself, up to terms that are very small, second order in the deviation \(\delta x(t)\) away from the classical trajectory! So indeed, for a trajectory to "solve the classical equations of motion" and for it "to constructively interfere with the close neighbors" are the very same thing, and that is why the family of trajectories that are close to the classical one "win" in the classical limit!

If you didn't really understand the path integral or quantum mechanics or if you were bad at deriving similar explanations, you could derive the classical limit by reducing Feynman's approach to the Schrödinger or Heisenberg picture, and to the arguments that illuminate the classical limits in those pictures. But you may see what happens in the classical limit using the Feynman picture itself and nothing else – and this perspective on the classical limit is truly elegant. Feynman has arguably made the Lagrangians great again... I actually mean "great for the first time in the history". Before Feynman, Lagrangians and actions (and Poisson brackets etc.) weren't really too cool, at least not to those who overlooked their elegance that included a prophetic flavor of the future quantum mechanical paradigm. Maybe Lagrange, Hamilton, and all these men "saw much of the beauty of QM" in their otherwise contrived actions and Poisson brackets (which were arguably "unnecessarily abstract and more complicated than Newton's equations").

Feynman has derived his diagrams, the leading method to compute all amplitudes in quantum field theory, directly from (also his) Feynman's path integral. Freeman Dyson (who died 368 days ago, after a fall) gave us another derivation, one built upon the operator approach. Let me start with that Dyson's argument because it's "conventional" in some sense. A non-PhD Dyson who should have been the "real maverick" actually gave us the derivation that most students consider "more mainstream" although this label is debatable. Well, in the operator approach, the unitary evolution matrix is a product of many factors of the form \((1+i\cdot \delta t \cdot H)\) (which represent the evolution over the infinitesimal time \(\delta t\)) and if the factors were \(c\)-numbers, the product could be written as the exponential of the sum (or "of the integral"). It is still possible when they don't commute but we need to add the "time-ordering operator". So the \(S\)-matrix is something like\[ S_{\rm matrix} = T\,\exp\zav{\int dt (-i) H(t)} \] Well, he did this exact thing but in the Dirac interaction picture. What does it mean? Following Dirac's trick, Dyson divided the Hamiltonian to the at most quadratic pieces \(H_{\rm free}\); and cubic and higher-order pieces \(H_{\rm int}\). The former was used to evolve the operators which is easy because with a quadratic Hamiltonian, it's just some multi-dimensional harmonic oscillator in the Heisenberg picture. The latter parts, \(H_{\rm int}\), were treated as perturbations. They produced the vertices and each field participating had to be paired with another one (from another vertex or the external states) and these pairs of field operators produced the propagators (as the time-ordered expectation values of products of the fields in the Heisenberg picture).

Again, some students imagine that the operator approach is the "normal thing" and the Feynman path integral has to be reduced to it. But that's not what Feynman did when he derived the Feynman diagrams for the first time. Instead, he really learned how to compute infinite-dimensional Gaussian integrals (the irrelevant but pretty square roots of \(\pi\) along with some determinants of the matrices hidden inside the free Hamiltonian appear in the result). And the interaction (cubic and higher) terms are decomposed to fields again and each field may be obtained by differentiating the original free, Gaussian integral with respect to \(J\) when \(J\phi\) source terms are added. One may see that the pairings and propagators are produced as well and the propagators arise as path integrals with two insertions (and therefore emerge from the inverse matrix to the matrix that appears in \({\mathcal L}_{\rm free}\)).

It is normal to spend weeks with learning the derivation of the Feynman diagrams, according to either picture. As a student of quantum field theory, you should know that this is a rare result. There aren't many things that are as important in quantum field theory (at least the practical one) as Feynman diagrams and their derivation. So just invest some time. If you can, learn Feynman's as well as Dyson's derivation. Understand how to translate between the two; why they are physically equivalent (at least in simple enough cases).

A funny thing is that the scattering amplitudes etc. may also be written as a sum of complex numbers that are encoded in the Feynman diagrams. This general Ansatz is the same as the Ansatz of the path integral itself: the scattering amplitude is a sum (or integral) of many complex numbers that represent different histories and that quantum (complex) interfere. The histories in Feynman diagrams look like particles that propagate (like in non-relativistic quantum mechanics of point-like particles; and that's why the QFT and non-relativistic QM notions of "propagators" end up being almost the same things, despite the "different" origin) and that sometimes split and join (at the vertices coming from the higher-order, interaction terms in the action). It is not just an analogy. By construction (taking Feynman's derivations of the diagrams from the path integral that I sketched above), the diagrams must be "the same thing", as Adam Lantos happily observed in the comments under the Quora answer.

The scattering amplitude is a sum (well, integral) over all histories which always includes the \(\exp(iS/\hbar)\) factor. But it also depends on the initial and final states, through the description of the particle species and their momenta, for example. But we do have a sum (integral) of infinitely many complex terms and these terms may be reduced to a different sum where "much of the path integral has already been calculated explicitly". It's only the fields at specific points that just participated in a Feynman vertex that create an "exception" and the Feynman diagrams simply reorganize the path integral over all histories according to all these exceptional points and fields where some virtual field quanta are created or destroyed!

In particular, it is still true that just like the path integral had to integrate over all histories and each of them has taken place in some deep sense, the calculation of the amplitudes using Feynman diagrams is also a sum over many "packaged histories" – they are given by the diagrams that look like histories of point-like particles including their mergers and splits, not by the field configurations \(\phi_j(x,y,z,t)\) – and all these diagrams have occurred in between the initial and final state again. Their constructive interference still helps them to give the process a higher probability although here, the role of the quantum interference could be said to be a bit smaller than in the \(\exp(iS/\hbar)\) calculation because unlike the \(\exp(iS/\hbar)\) terms, the Feynman diagrams may be larger or smaller which is why the absolute value finally matters, too, and maybe it matters a bit "more" than the phases. Simpler Feynman diagrams are generally "larger" because they are proportional to "smaller" powers of the small (and therefore "suppressive") coupling constants.

But it's important to realize that the sum of Feynman diagrams is still a reorganized sum over histories and all of these histories appear "simultaneously". If someone asks you whether a virtual particle pair or two were created in between the initial and final observations, you must say that you don't know. All the shapes of the Feynman diagrams with all the momenta of the virtual particles have contributed to the probability amplitude of the process – but you could only see one particular outcome, a final state, and its probability was the only thing that was calculable in quantum mechanics. The observables describing the final state were the only observable characteristic of the process. The intermediate history was hidden under George Bush's (or Bill Clinton's?) "Don't Ask, Don't Tell" mantra. (That mantra is being replaced with "Ask, tell, whine, cancel, and fudge others in unexpected combinations".)

Of course, in some cases, you may obtain a final state that is extremely likely to emerge from one particular Feynman diagram because the momenta match, they have a very special property etc. In that case, it is somewhat more reasonable to say that "the single diagram with the right fingerprint" is what has "made" this process end with the particular final state. But you can never be sure because all the intermediate states were "possible". How many virtual particles existed at \(t=0\), in the middle of the intermediate evolution of a scattering process? Well, all the numbers that are allowed according to a Feynman diagrams contributed. In effect, you may be asking about the wave function(al) of the system at \(t=0\) and indeed, it was almost always a "very complex superposition that included almost all possible terms with a nonzero complex coefficient".

However, I also need to emphasize that the notion of the "superposition of localized pieces" is only possible if you talk about the wave function at some given time (or, at least marginally, when the period of time under consideration is localized to some extent). If you talk about the history at all intermediate times, the "wave function" describing the histories that contributed is not localized (somewhat similarly to dark energy which is also not as localized as matter), as I said at the very beginning. Instead, every single history contributes with the weight \(\exp(iS/\hbar)\) which always has the same absolute value and the phase is the only thing that is affected by the initial and final conditions.

Feynman's path integral approach resembles modern art. It was a great contribution of the "second generation that did quantum mechanics". The zeroth generation was Niels Bohr who was born in the 1880s and he only invented the pre-quantum mechanics (of hydrogen atom) and was the spiritual father of the first generation (where Schrödinger didn't belong; he was an early anti-quantum warrior and he was born in 1887, about 0.2th generation, anyway). The first generation quantum mechanics dudes were overwhelmingly born between 1900 and 1902: Pauli 1900, Heisenberg 1901, Dirac 1902, Jordan 1902... OK, Max Born was born in 1882.

The path integral "reframing" of quantum mechanics due to the *1918 Feynman (yes, as old as Czechoslovakia) was much more original than the switching between Schrödinger, Heisenberg, and Dirac interaction pictures. It is really conceptually new although the theory isn't physically new. Feynman modestly downplayed this mini-revolution – a very decent attitude relatively to the anti-quantum hacks who liked to be presented as revolutionaries when they promoted another "reinterpretation" of quantum mechanics that was actually conceptually or physically erroneous. But Feynman really did appreciate that he was describing the same theory and all the philosophical aspects of quantum mechanics were explained by him in the path integral language, too. You can't say that some particular histories are real and others are not; you can't measure complementary observables at the same moment; it is really meaningless to ask "what precisely happened and didn't happen in between" the observations; the quantum phases are crucial for the first time in the history of probability.

In either picture, quantum mechanics is brilliant, perfectly internally consistent, and compatible with all the observations. But those positive traits only exist if you accept the qualitatively new philosophical axioms of quantum mechanics. Everyone who insists on squeezing quantum mechanics into a classical physics straitjacket is guarantee to remain a 17th century moron who will always overlook the beauty, elegance, and new "outside the box" solutions that quantum mechanics has irreversibly brought to physics.

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