Don't mess with the path integral

Tonight, the TV Prima Cool (Czechia) is going to air The Codpiece Topology, the episode 2x02 of The Big Bang Theory (in Czech dubbing).

It was the first episode I have ever watched (but later, I retroactively watched all the episodes, including the first season, usually several times) and because of this scene in which Sheldon graciously overlooks the fact that Leonard's new and temporary girlfriend is an arrogant, sub-par scientist who actually believes that loop quantum gravity better unites quantum mechanics with general relativity than does string theory, people started to talk about Sheldon Cooper's character being based on your humble correspondent.




Meanwhile, in 1986, Texas began to fight against litter on the roads. "Don't Mess With Texas" became their slogan. What does it have to do with the sub-par scientists and with my slogan, "Don't Mess With the Path Integral"? And what is the difference between the two situations? :-)

Well, let me begin with the main differences. The Texan littering was mostly done by young males but the analogous contamination of the concept of the path integral is done both by males and females - and in fact, females are relatively overrepresented in this business. Leslie Winkle is not quite an exception. ;-)

Why the path integral works

Great physicists such as Paul Dirac asked the question whether the old and pretty concept of an action that should be extremized - an elegant way to formulate the laws of classical physics - can play a role in quantum mechanics. The answer turned out to be Yes. Dirac, who previously proved the equivalence of Schrödinger's wave mechanics and Heisenberg's matrix mechanics, wasn't too far from finding the "third" major computational framework for quantum mechanics.

But it turned out that the world actually needed a younger, brilliant scientist with a huge experience with particular calculations to complete the picture. As you know, it was Richard Feynman who did it.



Before Richard Feynman, municipal attempts to make particles and vehicles move along several trajectories at the same moments ended in this way.

The probability (or its density) that the initial state "A" evolves into the final state "B" is calculated as the squared absolute value of the corresponding complex probability amplitude. This amplitude may be obtained by solving Schrödinger's equation; analogous and physically equivalent quantities may be extracted from Heisenberg's equations, too.

However, instead of writing a long algorithm to deal with the question, Richard Feynman wrote down the explicit answer. You don't have to "solve" anything. Instead, here is the result:

c(A to B) = integral (over histories from A to B) exp(i.S/hbar)

In some sense, quantum physics became simpler than classical physics because all the answers - to all questions because, according to quantum mechanics, all questions can be reduced to similar "A to B" probabilities - can be answered by a compact formula. In classical physics, you also have to talk a lot if you want to teach someone how to calculate the answers: "find a solution to this differential equation, and then do this and that". In quantum mechanics, you just say "shut and and calculate the integral that depends on A, B, and your particular questions in an explicit way".

To upgrade Feynman's formula from words to actions, you need to integrate over all trajectories in spacetime that begin at the configuration (in position space) "A" at time "t(A)" and end at "B" at "t(B)". Each configuration is added with weight "exp(i.S/hbar)" where S is the classical action - integrated Lagrangian - corresponding to the given history. And "hbar" is the reduced Planck's constant.

It works but many people - including people often considered to be physicists - completely misunderstand the necessary yet somewhat counter-intuitive conditions and subtle mechanisms that make it work. In this text, we will look at a couple of them - and I will also lightly mention the ways how these fundamental features are being messed with in the "discrete" approaches to quantum gravity, among various other memes whose goal is to rape and distort the basic principles of physics.

There are several features of the path integral that may look counterintuitive and superficially contrived if not harmful. But once you understand how quantum mechanics works, you will see that they're not only harmless; they're in fact critical for the consistency of quantum mechanics in this form.

Complexity of the integrand

First, a layman could expect that the integrand could (or should) be real. Why the hell does Feynman integrate some bizarre complex exponential? Couldn't (or shouldn't) he include, for example, an integrand equal to "exp(-S/hbar)" instead?

There has already been a text explaining why complex numbers are fundamental in physics on this blog but it didn't say anything substantial about Feynman's path integral. Feynman's approach to quantum mechanics makes the need for complex numbers particularly transparent.

It is not hard to see that you would get no interference etc. if there were no oscillating functions. But you could still think that the integrand could be "cos(S/hbar)", among other possible choices. Well, you would be wrong. The complex exponential is totally essential and unique.

If you calculate the evolution over an infinitesimal period of time, Feynman's path integral effectively tells you that the initial position is sharply defined. The uncertainty principle implies that the position is completely undetermined, so it must be possible for the particle to easily get to any point of space just a moment later.

Of course, you may calculate what happens with the delta-function initial wave function after a tiny moment of time according to Schrödinger's equation, and you will confirm the result obtained from Feynman's path integral. The evolution over finite intervals of time can be simply obtained by adding many copies of the infinitesimal evolution - and you obtain the integral over all histories as a result.

That's one way to get to Feynman's result but you shouldn't imagine it is the only way or "the canonical way". The result is more important than "the way".

The non-vanishing of the absolute value of the integrand

One can ctually prove much more about Feynman's path integrals than what we have mentioned so far: one obtains the same probability amplitudes from the path integral as he does from the Hamiltonian, operator approaches if the action - the integrated Lagrangian - if Feynman's exponent is related to the Hamiltonian by the usual relations we have known in classical physics.

(Recall that "H" is equal to "p_i v_i - L" where "p" are momenta, "v" are velocities, and the product is summed over "i".)

However, in the operator approach, the Hermiticity of the Hamiltonian is what is needed for the evolution operators to be unitary (the exponentials of "i" times Hermitean operators are the only systematic formulae to produce unitary operators) which is needed for the conservation of probabilities. It is required to guarantee that the total probability of all possible outcomes is still equal to one.

What does the unitarity - and the Hermiticity of the Hamiltonian - mean in the language of the path integrals?

Well, if you think mathematically for a while, and concentrated mathematical thought is what is needed for all such questions, you will easily find the answer. What the Hermiticity of the Hamiltonian translates to is the reality of the Lagrangian - or reality of the action.

But in this proof, the action still comes with the factor of "i" in the exponent of the path integral, much like the Hamiltonian enters Schrödinger's equation with an "i" in it, too. These two "i" factors are linked to one another, of course. (This "i" disappears in both approaches if you switch to the imaginary time but then the connection with the observations in real time becomes nontrivial and only accessible through analytical continuation.)

So the reality of the action - which is needed to conserve the probabilities - actually means that the exponent in the path integral has to be pure imaginary! And the exponential itself must therefore be a phase - a complex number whose absolute value is equal to one (or, to allow for a better normalization, is a constant independent of the history).

So up to the phase, each history must actually contribute "equally strongly" to Feynman's path integrals, otherwise the probabilities wouldn't add up to 100 percent! This is an elementary fact about the whole method due to Feynman but I claim that already this very point is misunderstood by those who try to invent ad hoc rules. Why?

Inclusion of all histories without censorship

Well, they misunderstand this elementary point because they often like to censor subsets of the possible histories. What does it mean for them to declare that some histories are not allowed to contribute to the path integral at all?

It's simple. It means to set the integrand to zero for these histories - or trajectories. You can also set the integrand to zero if you redefine the action in the exponent to be "S=i.infinity" for the banned histories. If you multiply this action by "i/hbar" and exponentiate the product, you will get "exp(-infinity)" which is equal to zero.

But if you set the action for some histories to "i.infinity", the action isn't real anymore. And because of the arguments above, it's equivalent to a violation of the Hermiticity of the Hamiltonian. You may think that for some "faraway" undesirable histories, physics won't notice. But you're wrong: if your censorship itself was supposed to play a physical role, physics will notice and collapse. If you mess with the path integral in this way, the amplitudes will no longer be unitary. The probabilities will not add up to 100%. It's as simple as that.

Moreover, one can see the inconsistency of the censored histories in many other ways. If you try to remove some histories based on a global criterion, you will also violate the fact that the evolution operator between moments "t(A)" and "t(C)" can be written as the matrix product of the evolution between "t(A)" and "t(B)" and between "t(B)" and "t(C)".

These two operators are obtained by integrating over all "A-B" histories and all "B-C" histories, so their product must clearly depend on all histories between "A" and "C". For any path integral that would remove some histories according to their properties in the whole "A-C" interval - i.e. that would be censoring histories according to their correlated properties in both "A-B" and "B-C" intervals - the transitivity of the resulting evolution operators would fail, too. Waiting for 2 seconds wouldn't be the same thing as waiting for 1 second and another second. It would be bad.

This problem comes on top of the problem with unitarity. They're related problems but I surely don't claim that they're the "only" problems you can find in a censored version of the path integral. Quite on the contrary, you may probably find many more related inconsistencies.

Inclusion of non-differentiable histories

Another property of the path integrals is counterintuitive for the beginners: a typical trajectory that contributes to the path integral - even if you switch to the Euclidean spacetime by the Wick rotation which changes "exp(iS/hbar)" to "exp(-S_E/hbar)" in order to make the integrand more convergent - looks like Brownian motion, a random walk.

It is non-differentiable almost everywhere: the derivative diverges everywhere. It looks nothing like a smooth history. It looks nothing like the classical trajectories that the particle could follow in classical physics. It looks much like the chaotic temperature graphs or the graphs of the spot prices on Dow Jones. Again, the laymen - including the laymen who are paid as physicists - usually dislike this fact. They would prefer the smooth trajectories to dominate the path integral.

But once again, this fact is not a disease of the path integral. It is its fundamental virtue. If the path integral could be defined purely in terms of smooth trajectories or histories, it could never be consistent with the uncertainty principle. Why? Well, such an integral would still be a mixture of trajectories that have a well-defined position and momentum at each moment of time.

However, if the trajectory looks like a random walk, this undesirable conclusion is removed. You may calculate the commutator "xp-px" by inserting the position and momentum operators along the history in two different orders, and because the typical trajectories are behaving so violently, you may actually prove that the commutator is nonzero (it is "i.hbar"): the time ordering in which you insert "x" and "p" to two nearby moments matters.

This non-vanishing of the integrand for any trajectory is not a disease of quantum mechanics: it is its fundamental property that is actually responsible for many of its - experimentally verified - qualitative features. For example, quantum tunneling exists and means that a particle must always have a chance to penetrate through a finite barrier as long as the final state behind the barrier is energetically accessible (and compatible with the exact conservation laws).

This ability of any quantum system to "tunnell" is another universal fact - and it is another reason why you are never allowed to censor the trajectories that look too counter-intuitive to you; in fact, these trajectories support the "majority" of the path integral.

I don't want to discuss about the crackpot theories of physics in detail because they don't deserve it. But be sure that all these Causal Dynamical Triangulations etc. are trying to fix a problem that the authors understand but they always create a problem that the authors don't understand - by censoring the path integral (that's what the word "causal" means) and by other illegitimate interventions into the very basic structure of Feynman's framework.

The relativistic limits on speed are taken care of automatically

Another wrong expectation that a beginner could have - and usually has - is that if you allow the summation over all trajectories of a particle, the typical particles will move faster than light most of the time and this will automatically result in a violation of the special theory of relativity. So an overzealous physicist-beginner could argue that the path integral needs to be "regulated" in the political sense. We must manually prevent the particle from moving faster than light, right? That's how the newbie could imagine the propagators in quantum field theory to be evaluated.

However, this expectation is completely incorrect, too. While most of the histories that contribute to the path integral contain points or features of fields that are moving superluminally most of the time, the resulting physical predictions will actually be fully compatible with relativity as long as the action you started with is relativistic.

This requires some calculations and cancellations (between particles and antiparticles, among other things) but if you do it right, you will see that I am right, too. The physically meaningful quantities will end up being consistent with relativity even thought the sum over mostly superluminal histories of point-like particles underlies all the propagators in any Feynman diagram.

No "regulation" of the violent behavior of the path integral is needed. Quite on the contrary. Any "intervention" into the path integral that would drop some histories that fail to obey certain inequalities - that you incorrectly assume must be imposed on the individual basis - will result in a violation of the consistency rules such as the conservation of probabilities.

Relativity only implies and requires that the ultimate testable predictions prohibit the genuine information from propagating faster than light. But the intermediate steps - and the individual histories included in Feynman's sum - can never be harassed by similar conditions.

For a consistent local quantum theory described by a path integral, the only thing you may "invent" is the action (which must satisfy additional conditions). Everything else is given by consistency. In particular, all histories have to be summed over with the "exp(i.S/hbar)" weight.

The right classical limit must always be formally manifest

While the path integral is completely dominated by non-smooth histories, any quantum mechanical theory must actually reduce to the classical theory in the appropriate classical limit. In all known formulations of quantum mechanics, the reason is manifest. Feynman's approach is no exception. In fact, it is extremely manifest in this approach.

The reason why Feynman's path integral reduces to the right classical limit is that the phases "exp(i.S/hbar)" are random numbers "spinning" around the origin of the complex plane - if you deform the history just a little bit, the action changes - and their contributions largely cancel among nearby histories.

The only exception are histories such that all nearby histories have pretty much the same phase - i.e. the same action "S" - so that this cancellation doesn't take place. When is the action "S" of a history much closer to the action "S" of all nearby histories than "generically"? It happens if "S" has an extremum at the given point - the given history. Then all the first derivatives vanish and the graph of "S" is locally "horizontal" in all the directions of the infinite-dimensional space of histories.

But "S" is extremized - or "stationary" - exactly if the given history solves the classical equations of motion. That's how the action - the integrated Lagrangian - is used in classical physics. So if you average the transition amplitudes over a reasonable region in space so that you calculate the evolution of a small wave packet and not just a strict delta-function, the existence of the cancellation will neutralize the contributions of all trajectories that are not classical solutions. And only the trajectories in the vicinity of the classical solution will matter. That's also why the particle (or another physical system) will move along these classical trajectories.

To carefully check that this is the case in a given quantum mechanical theory formulated via the path integral, you may need to quantify lots of prefactors and check what's happening with the regulators if you divide the time into small pieces, and so on. However, this complexity of the calculation shouldn't distract you from a key fact that the condition actually has to be satisfied exactly!

If you build the quantum mechanical path integral from its classical limit because the limit is what you know empirically to be satisfied in Nature, it's important that the classical limit is actually reproduced. And the simple mechanism above is the only possible explanation why the classical limit comes out correctly.

If you have a theory in which the argument above cannot be made precise - if you have good reasons to think that the formal argument will be spoiled by many new history-dependent factors that may actually highlight different histories than the classical solutions, then your theory is simply screwed. It's dead and you should abandon it immediately.

Obviously, this is the case of all the discrete approaches to quantum gravity, too. Even if you pretend that you don't understand the condition of the unitarity and you censor subsets of the "most atrocious" histories in an ad hoc way, your formalism will still fail to reduce to the right classical limit simply because the histories that don't solve the classical equations may be "amplified" by lots of combinatorial factors that are characteristic for your theory.

Even if you argue that you don't exactly know how the "tetrahedrons" etc. should look like in your childish theory of spacetime, you must admit that some combinatorial structure is the very point of this philosophy. And this combinatorial structure implies that very different histories will come with hugely different combinatorial factors added in front of "exp(i.S/hbar)". This fact will completely damage the condition that the trajectories with the extremal "S" dominate the path integral - and this disease of your picture doesn't depend on any details about the polyhedrons you want to use: this disease directly follows from the "combinatorial" philosophy! So the philosophy itself is ruled out.

There may be e.g. lots of ways how to glue simplices and tetrahedrons in Causal Dynamical Triangulations in arrangements that don't resemble the classical solutions in any way - and don't have any reason to - and this combinatorial "advantage" will imply that your Causally Dynamically Triangulated theory cannot produce the right classical limit, not even after any finite sequence of "censorships". The path integral will always be dominated by the sickest possible histories that you haven't banned (yet).

The fundamental assumption of these theories - the assumption that the spacetime is discrete - is shown to be wrong. There are lots of Leslie Winkles in that field who are as blinded as Islamic fundamentalists and who simply can't imagine that their basic assumption about the reality could be falsified by a simple and indisputable argument. But it can and it has been. What they spent their lives with is complete nonsense.

The right way to deal with the path integral

Once again, the right way to deal with the path integral is to start with a classical action reflecting the known empirical data - or an action that can be guessed to lead to a renormalizable and/or otherwise interesting theory - and simply sum over all histories without any attempt to mess with the path integral. Whatever the path integral will tell us is a true prediction of the quantum theory.

The path integral will contain integrals over topologically nontrivial contributions. And indeed, they exist and influence the physical phenomena: we talk about instantons, sphalerons, and other histories (and their vicinities).

Also, if the topology of the "Universe" in your path integral seems to be variable, it is because it is variable. For example, you may compute the path integral for 2-dimensional theories with a metric tensor - i.e. theories of two-dimensional gravity. The path integral seemingly tells you that you should include genus g Riemann surfaces with an arbitrary number of holes as well.

You may have some preconceptions that you don't like it and it must surely be wrong. Except that it is completely and inevitably correct. You should shut up and calculate. In the case of the theories with two-dimensional gravity, the different topologies of the world sheet will simply generate loop corrections for string theory which is what you will converge to if you deal with the two-dimensional gravitating theories properly, too.

The path integral is always smarter than you - and it is zillions times smarter than all the advocates of discrete physical theories combined. At any rate, the lesson is, once again:

Don't mess with the path integral.

If you think that you're smarter than a physical theory and you have the credentials to censor or modify the path integral according to your preconceptions, then it is because you are, much like the left-wing politicians and ideologues, an arrogant imbecile who hasn't even understood the basics of quantum mechanics as sketched in this blog entry - not because you have a good point.

And that's the memo.