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Why isn't the path integral rigorous?

+ 8 like - 0 dislike
3014 views

I've recently been reading Path Integrals and Quantum Processes by Mark Swanson; it's an excellent and pedagogical introduction to the Path Integral formulation. He derives the path integral and shows it to be: $$\int_{q_a}^{q_b} \mathcal{D}p\mathcal{D}q\exp\{\frac{i}{\hbar}\int_{t_a}^{t_b} \mathcal{L}(p, q)\}$$

This is clear to me. He then likens it to a discrete sum $$\sum_\limits{\text{paths}}\exp\left(\frac{iS}{\hbar}\right)$$ where $S$ is the action functional of a particular path.

Now, this is where I get confused. He claims that, because some of these paths are discontinuous or non-differentiable and that these "un-mathematical"1 paths cannot be disregarded, the sum is not mathematically rigorous, and, thus, that the transition amplitude described by the path integral is not rigorous either. Please correct me if I am incorrect here.

Furthermore, he claims that this can be alleviated through the development of a suitable measure. There are two things that I don't understand about this. First, why isn't the integral rigorous? Though some of the paths might be difficult to handle mathematically, they aren't explicitly mentioned at all in the integral. Why isn't the answer that it spits out rigorous? And, second, why would a measure fix this problem?


1 Note: this is not the term he uses

This post imported from StackExchange Physics at 2015-05-13 18:55 (UTC), posted by SE-user jimmy
asked Mar 11, 2015 in Theoretical Physics by jimmy (40 points) [ no revision ]
Short answer: To define an integral rigorously, it's not enough to just say "and now take the limit $N \to \infty$". You need to prove that your discrete sum converges to something, and that it doesn't matter how you take the limit.

This post imported from StackExchange Physics at 2015-05-13 18:55 (UTC), posted by SE-user Javier
@Javier Badia does this have to do with non-differentiable paths or is it a separate issue?

This post imported from StackExchange Physics at 2015-05-13 18:55 (UTC), posted by SE-user jimmy
Can't one make everything work with proper regularization, and doesn't this regularization allow all the cases actually relevant to physics (as opposed to all possible mathematical corner cases)?

This post imported from StackExchange Physics at 2015-05-13 18:55 (UTC), posted by SE-user DanielSank
It's 'excellent and pedagogical' compared to what other presentation?

This post imported from StackExchange Physics at 2015-05-13 18:55 (UTC), posted by SE-user NikolajK
It is indeed folklore that path integral is not rigorous mathematically, or more precisely, the rigorous maths has not yet been rigorously developed. This is typical in physics. But the real problem is that, many people do not know they are doing handing waving when they are doing it.

This post imported from StackExchange Physics at 2015-05-13 18:55 (UTC), posted by SE-user Jiang-min Zhang
@NikolajK just in general. it is my first introduction to path integrals, but I am finding that the book is not too difficult to follow

This post imported from StackExchange Physics at 2015-05-13 18:55 (UTC), posted by SE-user jimmy

See also this discussion.

3 Answers

+ 6 like - 0 dislike

There are several points:

  • The first is that for usual self-adjoint Hamiltonians of the form $H=-\Delta +V(x)$, with a common densely defined domain (and I am being very pedantic here mathematically, you may just ignore that remark) the limit process is well defined and it gives a meaning to the formal expression

    $\int_{q_a}^{q_b} \mathcal{D}p\mathcal{D}q\exp\{\frac{i}{\hbar}\int_{t_a}^{t_b} \mathcal{L}(p, q)\}$

    by means of trotter product formula and the corresponding limit of discrete sums. So the object has most of the time meaning, as long as we see it as a limit. Nevertheless, it would be suitable to give a more direct mathematical interpretation as a true integral on paths. This would allow for generalizations and flexibility in its utilization.

  • It turns out that a suitable notion of measure on the space of paths can be given, using stochastic processes such as brownian motion (there is a whole branch of probability theory that deals with such stochastic integration, called Itô integral). To relate this notion with our situation at hand there is however a necessary modification to make: the factor $-it$ in the quantum evolution has to be replaced by $-\tau$ (i.e. it is necessary to pass to "imaginary time"). This enables to single out the correct gaussian factors that come now from the free part of the Hamiltonian, and to recognize the correct Wiener measure on the space of paths. On a mathematical standpoint, the rotation back to real time is possible only in few special situations, nevertheless this procedure gives a satisfying way to mathematically define euclidean time path integrals of quantum mechanics and field theory (at least the free ones, and also in some interacting case). There are recent works of very renowned mathematicians on this context, most notably the work of the fields medal Martin Hairer (see e.g this paper and this one, or the recent work by A. Jaffe that gives an interesting overview; a more physical approach is given by Lorinczi, Gubinelli and Hiroshima among others).

  • The precise mathematical formulation of path integral in QM is called Feynman-Kac formula, and the precise statement is the following:

    Let $V$ be a real-valued function in $L^2(\mathbb{R}^3)+L^\infty(\mathbb{R}^3)$, $H=H_0+V$ where $H_0=-\Delta$ (the Laplacian). Then for any $f\in L^2(\mathbb{R}^3)$, for any $t\geq 0$: $$(e^{-tH}f)(x)=\int_\Omega f(\omega(t))e^{-\int_0^t V(\omega(s))ds}d\mu_x(\omega)\; ;$$ where $\Omega$ is the set of paths (with suitable endpoints, I don't want to give a rigorous definition), and $\mu_x$ is the corresponding Wiener measure w.r.t. $x\in\mathbb{R}^3$.

This post imported from StackExchange Physics at 2015-05-13 18:55 (UTC), posted by SE-user yuggib
answered Mar 11, 2015 by yuggib (360 points) [ no revision ]
thanks, great answer

This post imported from StackExchange Physics at 2015-05-13 18:56 (UTC), posted by SE-user jimmy
+ 1 like - 0 dislike

The Lagrangian involves derivatives but the differentiable functions have measure zero in any useful definition of a measure on a function space (for example the Wiener measure), so they would integrate to zero. This makes the introduction of the approximate path integral mathematically dubious.

The second dubious point is that the sum is highly oscillating and not absolutely convergent. Hence unlike for a Riemann integral, the sum doesn't have a sensible limiting value as you remove the discretization. The sum depends on how one orders the pieces and (as is well-known for many divergent alternating series) can take any desired value depending how you arrange the terms. Note that there is no mathematically natural ordering on the set of paths. Thus the recipe doesn't give a fixed result.

Therefore the informal introduction is only an illustration - an informal extrapolation of what can be made well-defined in finite dimensions (through treating $i$ as a variable and analytic continuation from real $i$ to $i=\sqrt{-1}$). In quantum field theory it is just used as a (very useful) formal tool to be used with caution since it may give wrong results, and only practice teaches what caution means.

Making the path integral well-defined in the setting discussed requires far more mathematical background and has been rigorously achieved only for quadratic Lagrangians in any dimension, and in dimension $<4$ for some special classes of nonquadratic ones. The mathematically simplest case is in $d=2$, where the construction of the measure (for real $i$), and the analytic continuation to $i=\sqrt{-1}$ are rigorously derived - it takes a whole book (Glimm and Jaffe) to prepare for it.

answered May 17, 2015 by Arnold Neumaier (12,355 points) [ revision history ]
edited May 17, 2015 by Arnold Neumaier
Most voted comments show all comments

The derivative term in the path integral $\dot{x}$ does not mean that the function $x(t)$ is differentiable. it means something else, namely $x(t+\epsilon)-x(t)\over \epsilon$ for the $\epsilon$ of your regularization. This quantity does not commute with $x(t)$ in the path integral, because $x(t+\epsilon) \dot{x} - x(t)\dot{x}$ is $(x(t+\epsilon)-x(t))^2\over \epsilon$ which is a fluctuating quantity which averages to 1 as a distribution in the standard quadartic path integrals.

This is not a problem in defining the path integral rigorously, because Feynman never expected that the typical $x(t)$ would be differentiable when writing down $\dot{x}$ in the path integral! He explicitly said it would be nondifferentiable, and he identified the non-differentiability as the path-integral origin of the Heisenberg commutation relation, whose imaginary time form I gave above.

There is never an expectation when you write a path integral that the differentiation operations are being applied to differentiable functions, when you are done with the continuum limit, the differentiation operators are applied in the distributional sense to distributions, the typical paths in the path integral are well defined as distributions in nearly all cases of interest. Further, the products of distributions in the path integral which have coinciding singularities are not interpreted as pure products either, these would not make sense, but rather as products in a regularization $\epsilon$, with subtractions which make them well defined in the limit of small $\epsilon$. The two ideas together, of operator products and distributional derivatives, give meaning to every term in the Lagrangian, without any a-priori assumptions on the character of the path, other than that it can be sampled by Monte-Carlo sampling (which is true numerically, and defines a probabilistic algorithm for making complete sense of the procedure).

The proper interpretation of the path integral is by a limiting procedure which introduces an $\epsilon$ and takes a limit at the end, i.e. a regularization and renormalization. This is also true in 0+1d quantum mechanics, as I tried to make clear using the noncommutativity of $\dot{x}$ and $x$. In 1d, there is no remaining problem with products in the path integral, other than this finite noncommutativity, and mathematicians have already defined a rigorous calculus, Ito calculus, for the imaginary time formulation of 0+1d. In field theory, there are also regularization issues with coinciding products, and the commutation relations expand to OPE relations, and you can't use the same calculus. This is also true for Levy field theory, where you replace the $\dot{x}$ term with the log of a Levy distribution between consecutive steps, to describe a particle making a Levy flight. This has a continuum limit also, but it is not described by Ito calculus, rather by a different calculus which has not been described by mathematicians.

The only issue with the path integral is how to take limits of statistical sampling for small lattices, or relaxing any other regulator. The limit is universal for small $\epsilon$, it doesn't depend on the discretization, just as the limits of discrete difference equations are universal in the limit of small $\epsilon$ and give you the differential equations of calculus. Even in calculus, it is difficult to prove convergence of arbitrary discretizations, the standard proofs of existence/uniqueness of differential equations iterate the integral form of the equation and don't bound the convergence of Runge-Kutta schemes rigorously, as the iterated integral equation already takes a limit implicitly inside the integral, and the convergence proof is made easier. You can't use such tricks in quantum field theory, except when you formulate the theory as an SDE (which is what Hairer does, and he iterates an integral equation to find his solutions).

The formal mathematics for describing the limits of measures to continuum measures is complicated by the difficulties mathematicians have for making a natural measure-friendly set theory, as current set theory makes any discussion of measure a minor nightmare, because you have to constantly keep in mind which sets are measurable and which are not when using all the classical theorems. This is not acceptable, and the better solution is to work in a universe where every subset of [0,1] is automatically measurable, and keep in mind that there are exceptions to the Hahn-Banach theorem, to the basis-existence theorem, the prime ideal theorem, in cases where the is an uncountable choice required in intermediate steps. Since all the actual mathematical applications require only countably many consecutive choices, the exceptions are isolated from mathematical practice by a brick wall, measure paradoxes hardly ever bother probabilists. When they have a construction which they prove is true for all real numbers, they will lift it to "the values" of random variables without worrying.

@RonMaimon The content of my previous comment was rather different from what you say. You are thinking about very elaborated constructions, that may be interesting nonetheless, I am talking about a very simple observation.

The Solovay model and the ZFC one are, concerning some assertions, in contradiction; in addition, some theorems of ZFC are not provable (or maybe false) in the Solovay model. This is a fact, and cannot be disputed. Another fact is that physical descriptions of the world do not use only probability theory and path integrals, but also very different mathematical constructions (and there is an equivalent, from a physicist point of view, formulation of QM and QFT that do not use path integrals so one may argue if that point of view is an unavoidable physical requirement, or just a convenient tool). I strongly believe that it would be very desirable that one could describe the physical world by means of a single coherent mathematical model, and not with two (or many, or infinite) that are in contradiction; and one has to choose the appropriate one each time to prove a result (in a setting where in the meantime other results are false).

Now, if you believe that the correct model should have all sets of reals measurable and so on, it is your choice and of course you have to be able to develop your theory in a consistent fashion, and provide the necessary results to describe the "world". Mathematicians do that using ZFC, and so are in contradiction with you, but not with themselves. And I am sure that many results of the probability theory that is done by mathematicians around the world using ZFC is correct, and takes into account the difficulties implied by the axiom of choice.

And if you prove a theorem in a model, that theorem is true, and that is by the very definition of proving a theorem. Given ZFC, and the Solovay model (or any other model you want), is up to physicists to choose the one that has the theorems that are better suited for their purposes, but they have to choose one nonetheless, for having multiple theories to describe one world is logically contradictory (and for human ontology as you call it, unacceptable). So my suggestion is: arguing on philosophy is very interesting as a hobby (and I enjoy discussing with you), but if you would like that your point of view become a valid mathematical alternative you should make axioms and prove new theorems (within your framework, and that means you have to redo most of the proofs). If the proof is mathematically acceptable, those theorems by themselves would be of interest in mathematics. To make the physicists (mathematical physicists at least, that use mathematical rigor) shift from the model of ZFC to your model, you have to prove a huge amount of theorems, namely all those that they use in physical modelling. If else, they would stay with the well-established and very powerful ZFC, for many many results are already available there.

@yuggib: I did not misinterpret your comment--- your new comment is saying the same thing "mathematicians are used to ZFC and don't want to change". This is an argument from tradition, and it fails, because tradition is failing terribly with measure theory.

In current mathematical practice, there are two completely incompatible conventions, both deeply embedded. It is conventional to assume the axiom of choice is universally true when doing set theory, and it is also conventional to simultaneously inconsistently assume, only when doing probability, that random variables can be spoken of as taking on values. This latter manner of speaking is so intuitive and convenient for describing arguments, that it is impossible to get probabilists to stop doing it, that's what they are doing in their heads to produce and internalize theorems, and that's how they speak about these theorems when they prove them. But the disconnect with set theory means that any such proof in probability cannot be made rigorous as it stands, but needs a relatively heavy slog to turn it into a statement of a different kind about sets, which does not at any point speak about random variables having actual values, because they can only be spoken of as members of a subuniverse of measurable sets. This procedure is extremely time consuming, it makes a disconnect between probability discussions and rigorous proofs, and because what you are doing in your head is completely different from what you are doing in the formal procedure, it is error prone, and can produce false proofs. The results are usually true anyway, because there is nothing really wrong with imagining random variables have values in a Solvay model.

In practice, it is very easy to get people to shift underlying frameworks, after you do a certain amount of work, because people don't use frameworks, they use theorems. The framework is like the operating system, you can invisibly insert a new one in when you can emulate the previous system, that is, when you include the usual system inside.

The political problem with Solovay's model, as you said, is that it is incompatible with some of the theorems mathematicians are used to, it doesn't immediately include the old system inside. This makes people wary, because they don't know what is true and what is false regarding the model anymore. But the old theorems are still valid in a more restricted sense, using dependent choice, and you need to keep track of this anyway, for probability.

But mathematicians would still like to know what the Hahn Banach theorem means in an expanded sense, or the prime ideal theorem. I am saying that you can interpret them in an expanded sense also, even in an imagined universe where all subcollections of R are measurable. But you need to be careful to talk about power-sets as incomplete collections. This is useful for forcing also, because right now, set theorists don't have a consistent single picture of the universe that they can work in, they use different pictures for different models of ZFC.

When you switch foundational stances, you don't have to reprove all theorems, you just have to show how the old system embeds in the new, and there should be no difficulty to embed the old in the new when you do things right. The Solovay model should have been close enough, but it hasn't been. I don't believe it is feasable to convert the arguments regarding path integration into theorems if it is not immediately apparent how to deal with infinite collections of random variables without contradiction. Saying "just pretend they don't have values" doesn't work for path integrals, because the arguments are too elaborate and require using properties of the distributions you produce in the limit. You can always limp along, but mathematics progresses not so much by finding better formal methods, but by making intuitive shorthand for methods that can be made formal, so that more complicated formal methods become easier to describe. I don't see any hope for the current probability formalism, as it will never allow you to speak about randomness naturally.

@RonMaimon What I am saying is that mathematicians are used to ZFC but they may change without much effort, for the mathematical results are theorems proved within a given logical system (that most of the time is ZFC but not always, the important thing is that it is clear what it is). The point is what physicists would like to do, which model they consider more fit to their needs to model reality.

And what you say about "embedding" theorems from one model to another, this is simply not true. Given a theorem of ZFC, it is true that a weaker version may hold (it is not assured, in the book about the axiom of choice by Herllich you can see explicit counterexamples) in ZF+(some weaker choice). There are theorems that no matter how weakened cannot be proved in ZF+(some weaker choice). And the Solovay model is not just ZF+DC+(large cardinal), but a model where some true statements of ZFC are false. There is obviously no hope of embedding them, for they are false! And it is very difficult to keep track of what is false in these "forcing models" that are in (partial) contradiction with ZFC: in fact, even if they have been introduced more than 50 years old, and people is well aware of them, but no "embedding rule" has been produced (while for example there is one in quantum logic and quantum set theory wrt ZFC, and that theory is far more recent). Finally, take into account that when dealing with advanced mathematics is already very difficult to understand if a theorem of ZFC is true in simple ZF+DC (the equivalence between a theorem and the axiom of choice or some of its weaker versions has been proved only in few instances, you can see many of them in the book I mentioned before); and I repeat that a given  theorem may be not only unprovable but false in the Solovay model.

To finish the discussion (at least by my side): I agree that these models and possibilities are intriguing and interesting; simply for now they are nothing more than a mere curiosity for mathematicians, and not so useful for physicists because so many important results are lacking in this context. And I am not the advocate of ZFC, first of all because it does not need me as an advocate, and also because the logical models are more a language to do proofs than entities by themselves in my opinion, so I don't think one is better than the others, but just different.

@RonMaimon: The problem is that once you change the interpretation as you want to, the old ZFC theorems no longer apply to your random variables, which are outside the protected subdomain in which the ZFC proofs remain valid. Thus you have to reprove every result you use on your random variables, since the ZFC proofs don't apply to them. This is a lot of work, and is (I guess) what @yuggib meant. One uses without thinking a huge number of facts that all become uncertain to any ZFC-based mathematician when using them on your random variables! Unless you can tell us where they are proved on your new basis.

So far all your aguments amount only to handwaving that there should be no problems in reproving the part needed outside. And as you can't refer to a source where all this is actually reproved for the instances you need to define and use your constructions in a real physical context, its just conjecture, not fact.

Most recent comments show all comments

That's exactly why there are two distinct theoretical communities in physics - the theoretical physicists and the mathematical physicists. 

The physicists you mention produce valuable results at the level of theoretical physics but only conjectures at the level of mathematical physics. I have nothing against doing theoretical physics at a formal nonrigorous manner, but it shouldn't be called rigorous before it isn't.

Conventions are a matter of social agreement, hence involve many people, not just one who is discontent with the existing conventions. If you want conventions to change you must write enough papers and books that demonstrate to the satisfaction of others in your target community that what you do is fully rigorous on the basis of the Solovay model. As mentioned already, it will probably mean 1000 pages rather than 20 paragraphs of handwaving. It is only this sort of insistence that changes conventions (and even then only in the long run) - continually writing papers that drive home the point, in a convincing way.

@ArnoldNeumaier: I do not accept that mathematical physics can operate with the current framework of probability. You simply have no idea how simple probability becomes in a universe where R is measurable, and once you see it, even a little, you can't go back. It's like unlearning calculus and going back to Riemann sums. The constructions in measurable universe bear more of a relation to the arguments of theoretical physicists than to the arguments of mathematical physicists. It's the mathematical physicists who are doing things all wrong in this case, not the theoretical physicists.

Many mathematicians, perhaps a large minority, perhaps a majority, are already dissatisfied with the inability to speak intuitively about probability in ZFC. This is reflected in the set theoretic paradoxes I linked, which you haven't read--- these are all conflicts with intuitive probability (for example, here: https://cornellmath.wordpress.com/2007/09/13/the-axiom-of-choice-is-wrong/ , related articles are "Axiom of Choice and predicting the future", and a linked mathoverflow question regarding an analogous even more counterintuitive issue with mathematicians unjustifiably guessing one of countably many real numbers in boxes put in front of them). These probability paradoxes are the main intuition conflict that led to issues with choice. You can read grumbling about it all over the literature. Conway complains about non-measurable sets in his book on Real Analysis. Connes complains about the fiction of non-measurable sets in his book about noncommutative geometry. Lebesgue complained about them in the 1930s until he died, and Cohen's forcing is accompanied by clear intuition straight from intuitive probabality. Solovay's paper by itself was nearly enough to completely change the consensus in 1972, there was a (small) movement to do it in the 1970s, and it didn't gather momentum, and Solovay's paper did not run to 1000 pages, it was a dozen or two relatively difficult pages (but they've been internalized and digested now, so that they're easy).

Why didn't the "revolution" complete? Why haven't all the measure theory books been rewritten so that every set is measurable? I believe for two reasons--- first, Solovay politically sold out a little, and explained in his introduction that "of course, the axiom of choice is true", and then gave some pieties to support conventional wisdom. He was not seeking a clean break with set-theoretic measure theory, like Lebesgue, Connes, Conway, and everyone else who knows intuitive probability. He was motivated simply by the desire to find a nifty model of ZF+DC, not by the desire to give a better model of "mathematical reality".

The other reason is that Solovay's model is not exactly a model of naive probability either. Sure, every subset of R is measurable, and the foundations of real analysis and functional analysis go through, so you can pick countably many numbers in [0,1], pick countable sequences of random variables and lift results from realizations, so that you can do the ordinary stuff physicists do for a path integral, but there are also intuitions that fail. Uncountable ordinals don't embed in R, the theory is still ZF underneath, so the intuition from intuitive probability that R is enormously big, bigger than any ordinal, is not preserved in Solovay's model, it can't be in any model of ZF.

For this same reason, preserving powerset axiom, his construction is extremely complicated to follow, it requires collapsing a whole universe to consist entirely of countable sets (this is a Levy collapse of an inaccessible to omega-1, it was explained to me recently what this collapse does), and this required a large-cardinal hypothesis. The large-cardinal is overkill for intuitive probability, you don't need it. You need it to make the powerset axiom true, because now the powersets and the uncountable ordinals are totally separate sequences (and you need powersets for the ordinals, and so on). This is the real headache--- staying consistent with powerset.

The results of mathematical physicists are unacceptably weak, and unacceptably obfuscated. I will repeat, there is nothing wrong with intuitive probability, it is all just fear mongering. There is nothing really wrong with the axiom of choice either. it's the axiom of powerset that is causing difficulty, and this axiom is used to define R as a set, and to define functions, and so on. The whole point of redoing the foundations is to make R a proper class, as advocated recently by Nik Weaver in his forcing book.

+ 0 like - 1 dislike

...He derives the path integral and shows it to be: $$\int_{q_a}^{q_b}\mathcal{D}p\mathcal{D}q\exp\{\frac{i}{\hbar}\int_{t_a}^{t_b} \mathcal{L}(p, q)\}$$

This is clear to me. He then likens it to a discrete sum $$\sum_\limits{\text{paths}}\exp\left(\frac{iS}{\hbar}\right)$$ where $S$ is the action functional of a particular path.

Now, this is where I get confused.

At this point I think it will be helpful to make an analogy with an ordinary Reimann integral (which gives the area under a curve).

The area A under a curve f(x) from x="a" to x="b" is approximately proportional to the sum $$ A\sim\sum_i f(x_i)\;, $$ where the $x_i$ are chosen to be spaced out from a to b, say in intervals of "h". The greater the number of $x_i$ we choose the better an approximation we get. However, we have to introduce a "measure" to make the sum converge sensibly. In the case of the Reimann integral that measure is just "h" itself. $$ A=\lim_{h\to 0}h\sum_i f(x_i)\;, $$

In analogy, in the path integral theory of quantum mechanics, we have the kernel "K" to go from "a" to "b" being proportional to the sum of paths $$ K\sim\sum_\limits{\text{paths}}\exp\left(\frac{iS_{\tt path}}{\hbar}\right) $$

In this case too, it makes no sense to just consider the sum alone, since it does not have a sensible limit as more and more paths are added. We need to introduce some measure to make the sum approach a sensible limit. We did this for the Reimann integral simply by multiplying by "h". But there is no such simple process in general for the path integral which involves a rather higher order of infinity of number of paths to contend with...

To quote Feynman and Hibbs: "Unfortunately, to define such a normalizating factor seems to be a very difficult problem and we do not know how to do it in general terms." --Path Integrals and Quantum Mechanics, p. 33

In the case of a free particle in one-dimension Feynman and Hibbs show that the normalization factor is $$ {({\frac{m}{2\pi i\hbar\epsilon}})}^{N/2};\, $$ where there are N steps of size $\epsilon$ from $t_a$ to $t_b$, and N-1 integrations over the intermediate points between $x_a$ and $x_b$.

Again, quoting from Feynman and Hibbs regarding these normalization measures: "...we do know how to give the definition for all situations which so far seem to have practical value."

So, that should make you feel better...

This post imported from StackExchange Physics at 2015-05-13 18:55 (UTC), posted by SE-user hft
answered Mar 11, 2015 by hft (-10 points) [ no revision ]

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