# Path integrals and foundations of quantum mechanics

1. Sep 2, 2011

### Demystifier

It is frequently stated that path integral formulation of quantum mechanics is equivalent to the more traditional canonical quantization.

However, I don't think it is really true. I claim that, unlike canonical quantization, path integral quantization is not self-sufficient. That's because the path-integral formulation itself does not contain a notion of a quantum state living in a Hilbert space, nor it contains any substitute for it. Instead, a self-sufficient formulation of quantum mechanics using path integrals must borrow the notion of quantum states from the canonical quantization.

For example, how would you derive violation of Bell inequalities from the path-integral formalism? I don't think you could do that.

My central claim is also reinforced by the fact that I can't remember that I ever seen a relevant paper on FOUNDATIONS of quantum mechanics based on path integrals. If you do foundations of quantum mechanics then states (especially entangled states) play a mayor role, and for such purposes the path integrals are not sufficient.

What do you think?

2. Sep 2, 2011

### xts

I may agree it is not complete picture (however I am not brave enough to discuss if entanglement cannot be expressed that way). I see path-integrals (in Feynman's interpretation) as an alternative formalism equivalent to canonical, giving just different philosophical view: like Lagrange's mechanics leads to different view than Newton's one.

3. Sep 2, 2011

### kof9595995

Interesting, I'll keep an eye on this post.

4. Sep 3, 2011

### tom.stoer

Demystifier is partially right. The PI is constructed from the Hamiltonian living in a Hilbert space (Feynman).

But of course there is nothing wrong with writing down a PI and say "this is the definition of the quantum theory" - as long as it works. I mean in the very end no "quantization" is a strict derivation b/c it has an incomplete - classical - starting point. But we do not have a rigorous "quantization" w/o using some classical expression.

So both ways
1) write down S - write down exp(iS) and integrate it
2) write down S - derive H - replace {.,.} with [.,.] and introduce the Hilbert space
are not self-sufficient.

In QM canonical quantization looks more fundamental, but looking at QFT I would say that it's exactly the other way round b/c only via S we can write down the correct symmetries; it's impossible to guess and write down the QCD Hamiltonian w/o using S!

So in the end it boils down to the question whether there are calculations which are impossible in principle (not only in practice) in the PI approach: Is it possible to show that e.g. Bell's inequalities cannot be derived and interpreted (!) via a PI? I mean something like a rigorous proof like the impossibility of squaring the circle using ruler-and-compass construction.

5. Sep 5, 2011

### Demystifier

Let me try to further refine my claim.

What are path integrals useful for? If you know the initial state, then path integrals can be used for calculating the state at an arbitrary later time. In QFT, this method may be even more effective than canonical methods, especially at t-> infinity.

However, to define the initial state itself in the first place, I don't think it can be done with path integrals alone.

6. Sep 5, 2011

### Demystifier

One additional note. Is it possible to calculate S-matrix elements in path-integral QFT without borrowing Hilbert states from canonical methods?

Most QFT textbooks using path integrals do indeed borrow Hilbert states from canonical methods. The only exception I am aware of is the beautiful textbook by A. Zee. Instead of particles defined as Hilbert states, he defines particles in terms of Schwinger sources. Even though this method involves a lot of hand-waving, it is probably sufficient for most practical purposes in particle physics. But can Schwinger sources be used to define entangled particles? I am afraid they can't.

7. Sep 5, 2011

### tom.stoer

Demystifier, I share your opinion that PI quantization is "not as fundamental" as the canonical formalism. But I disagree with your claim that canonical quantization is self-sufficient (b/c it borrows from classical mechanics). No "quantization" of a classical formulation can be self-sufficient simply b/c of the classical starting point.

8. Sep 5, 2011

### Prathyush

Can one write a path integral for a spin half particle without space time degrees of freedom. In specific i cant see if one can write an action principle that reduces to

$H = \sigma.B$

It seems to me that once cannot define a action given an arbitrary Hamiltonian. How ever i do agree that symmetries are transparent in the action formalism.

9. Sep 5, 2011

### tom.stoer

What would be your canonical conjugate variables q and p?

10. Sep 6, 2011

### Demystifier

I am not sure that I correctly understood you, so let me probe my understanding by an additional question. Is a theory consisted of a classical Lagrangian and a canonical method of quantization self-sufficient? (I would say it is.)

11. Sep 6, 2011

### Demystifier

q and p are not the only bases on which a path integral can be based. Another popular choice is coherent-state basis.

For a PI of spin, search also for
Ben Simons, Concepts in Theoretical Physics
which is a textbook on QFT which can be freely (and legally) downloaded from internet. I recommend this book for other reasons as well, because it presents QFT from a somewhat unusual point of view.

12. Sep 6, 2011

### tom.stoer

In general it isn't, simply b/c you have operator-ordering ambiguities. Consider a free particle constraint to a curved manifold; quantizing this theory in position-space results in a second-order differential operator corresponding to d²/dx²; usually one uses the Laplace-Beltrami operator Δg depending on the metric g on that manifold in order to get mathematically reasonable result. But there is no step in the canonical quantization procedure that tells you why exactly you have to use this special operator ordering. Something similar applies to the measure used in the inner product on the state of space as well.

So in order to have a self-complete canonical quantization it seems that you have to specify more than just the canonical variables. In QFT and especially QG (LQG) it seems that it is still not completely clear how to implement constraints (especially constraints resulting from gauge symmetries); for the diffeomorphism constraints it is well-known that it does not generate a constraint algebra with structure constants like fabc in SU(N) but a constraint algebra with structure functions depending on the canonical variables which gives rise to new operator ordering ambiguities. There is no standard way to resolve them, one always has to use "physical reasoning". Unfortunately wrong choices may generate gauge anomalies. afaik not even in LQC which is a theory of finitely many degrees of freedom (and which should therefore be free of problems generated by field theories) not all these issues are resolved.

I agree that a completely specified canonical qantization scheme which fixes all these ambiguities IS self-contained, but unfortunately most of these schemes would generate "physical nonsense". In order to identify the "physically reasonable" schemes one again needs "physical insight".

Last edited: Sep 6, 2011
13. Sep 6, 2011

### Fra

From my own inference-inspired perspective, I think the PI is much more intuitive. The actual PI as a computing an observer dependent expectation as weighted sum of over observer dependent distinguishable transitions from the prior state to a possible future state is a very clean abstraction of how an intrinsic inference works.

The foggy parts is

1) exactly how to "count" the possible transitions and what their relative weights are (problem of normalization and the problem of the choice of integration measure)

In inference this is the problem of "how to count" evidence. Ie. how to do impose a measure on the set of evidence, so you can defined negotiations.

2) how to combine the "possibilities" into one transition probability (the problem of quantum logic)

In inference, this is the problem of defining a single measure on the space of conjuctions of two other spaces that doesn't commute. This is needed to "combine evidence" that are not independent. INFORMATION about q is not independent of INFORMATION about p, for example.

But these problems, appearing natural in this picture are natural in an inference context. They all have a conceptual hook.

Actually the problem Tom often mentions, the nuisance that we always have to rely on a "classical input" in a somewhat ad hoc way (lagrangian or hamiltonian), is IMHO more easily attacked in the PI formalism.

This is because if you add a conjecture about "rationality" upon the "expectations" - seens as "counting evidence", then the information that comes from the classical input, is instead encoded in an evolved rationality condition. What this means in clear is that the action of the system when rational is a pure random walk, which renders ALL actions as entropic.

The intuitive idea is that if we let a group of INTERACTING players make RATIONAL random walks, there will be an equilibrium point where they develop nontrivial actions (that from the point of view of other observers is anything by random walks) that can be understood in terms of evolving entropic interactions.

This perspective is natural in the PI, nad why I like it.

I do not like the operator approaches at all, they strip out the intuitive picture for me. For me a measurement is not a "projection", a measurement is the backreaction of the black box in response to a random walk. The nontrivialiy arises from that fact that the backraction is not random, it feeds information back into the random walker which processes it and improves it's "random walking".

What is particularly completely lost in the mesaurement = projection picture is that the MASS of the observer is wiped away (ore rather assume infinite). This has exactly to do with the counting and normalization of the PI. So I think the "conceptual angle" is superior in the PI. In the inference picture, it's clear that no observer count to infinity. Instead the inference perspective itself imposes a cutoff, but which is not a regularization but have a physical explanaion.

Alot of the refined and algebraic approaches are nice and pure, but for me they do strip away some conceptual handles. In particular, they "purify" and axiomatize current theory to the point that it's harder to generalize.

/Fredrik

14. Sep 6, 2011

### Fra

My take on this is that the initial state is the "prior state".

The prior state (and the rational expectations implicit in it) is what defines the observer, so the question of explaining the only thing given seems strange. The only rational question is how to move forward (into the future), GIVEN the prior.

So the question becomes in my vivew: De we have a choice among infinte "prior structures" to make, each impliying an own version of "rational actions" (instead of classical inputs)? This would indeed render this picture unpredictable since in order to make a prediction we need to make a choice among infinitely many priors.

But what saves us is another thing. If we start considering the lowest possible complexity of an observer (thus constraining the state space of the priors) then we can reduce the space of possible priors to pretty much a single bit, or to the point where the options are small and finite. Then try to understand this picture, and later try to understand the interaction of such observers. Then add mass generation and ask how more complex interactions become "possible".... then add to this picture an evolution, where there is a selection takin place along with scaling up the observer mass. Quite possibly there is a unique limit here. It's understanding that this "limiting process" is not a regularization that is sometimes used ANYWAY in PI, but with much worse motivation, but actuall a physical evolution that connects to mass generation and encoding of interactions.

This is just some wild ideas (still I claim rational), but it seems the PI is very suitable and open for such things. It's easier to attach it into the framework.

/Fredrik

15. Sep 6, 2011

### tom.stoer

The question Demystifier asks is: if you write down something like Z = <s|s> using exp(iS) and define observables via Z[J], then how do you define this Z which depends on |s> w/o being able to write down |s> itself?

Usually you define Z using a Hamiltonian PI plus the Gaussian integration to eliminate the Dp; that results in a Lagrangian PI with Dq exp(iS). But the very starting point always is the Hamiltonian PI.

Now one could skip that (Feynman's) step and write down the Lagrangian PI immediately (even so it is unclear whether we would arrive at exp(iS) w/o knwowing how to do the Dp integration). This is what is usually done in QFT: nobody cares about deriving the Lagrangian PI via the Dq integration simply b/c constructing H is so awful. Anyway - it may be OK to define Z w/o using the Dpintegartion at all.

But it seems to be impossible to write Z w/o writing down |s>! But |s> is something that does not exist in the PI formalism! That's Demystifiers point!

16. Sep 6, 2011

### Fra

I'll get back later but just a short comment.

I'm not sure if part of the issue is definitions of PI or of I'm missing the point but a qiuck comment.

I think the notion of hilbert space gets replaces just be "microstructure" in the PI. Which is how I think of it anway. In principle the partition function implicitly encodes the microstructure of the observer (which is the equivalent of the hilbert space).

They way I think of things, I think in terms of a "generalized" system of microstructures (that doesn't ocmmute). This corresponds to a generalization of "stat mech". Transition probabilites are then seen as transitions between "sets of microstates" (observers state of konwledge).

So I propose that
hilbert space ~ system of microstructures (that are non-commuting)
state vector ~ the microstates in the set

For me the starting point is the set of microstructures; on which hte partition funciton is deifned. If we have this, we need to hilbert space.

I'll read again later tonigt... not sure if I missed the poitt still..

/Fredrik

17. Sep 6, 2011

### schieghoven

Interesting. Personally, I'm a fan of canonical quantization because I think it is easier to identify the underlying hypotheses of the theory.

I vaguely recall that some of the early proofs in non-abelian QFT were first constructed using path integrals -- Slavnov-Taylor identities & BRST renormalization, Gribov ambiguitiy. However, I don't know whether equivalent proofs can be constructed via canonical quant.

18. Sep 6, 2011

### schieghoven

Interesting. Personally, I'm a fan of canonical quantization because I think it is easier to identify the underlying hypotheses of the theory.

I vaguely recall that some of the early proofs in non-abelian QFT were first constructed using path integrals -- Slavnov-Taylor identities & BRST renormalization, Gribov ambiguitiy. However, I don't know whether equivalent proofs can be constructed via canonical quant.

19. Sep 6, 2011

### tom.stoer

There are a lot of papers regarding canonical approaches towards non-abelian QFTs, especially for low-energy physics (i.e. not scattering but spectra, form factors, ...) Gauge fixing can be defined rigorously, gauge-anomalies are absent (so this is something like Slavnov-Taylor identities & BRST), but of course you don't escape from Gribov ambiguities as these are due to the fibre bundle structure.

20. Sep 6, 2011

### tom.stoer

Let's make an example: usually in QM the path integral is constructed as a propagator K(xb,tb; xa,ta). This expression is derived from the matrix element <xb,tb | xa,ta> using insertions of the time evolution operator U(tb;ta). In the very end one gets an expression which does no longer contain any state vector but the usual Lagrangian PI in position space. It answers the question regarding the probability of a particle located at xa at time ta to propagate to xb at time tb.

Now let's consider a different question, namely regarding the probability for a particle to be in an energy eigenstate na at time ta to jump into a different energy eigenstate nb at time tb. We may e.g. look at a hydrogen atom which has been prepared in a certain state and we may detect the emitted photon in order to measure the probability.

In the Hilbert space formalism you simply write <nb,tb | na,ta> and you are ready for the calculation.

My question to you is: how do ask and answer this new question w/o writing down this matrix element? How do you ask and answer this question using nothing else but K(xb,tb; xa,ta)? No bras and kets allowed!