Exploring the Fundamental Postulates of QM: Are They Truly Ad-Hoc and Strange?

[HomogenousCow]

I find the fundamental postulates of QM very ad-hoc and strange.
Compare them to the fundamental postulates of special relativity, special relativity naturally arises out of classical electromagnetism and the equivalence of all inertial frames, while QM seems to come out of nowhere.

 

[tom.stoer]

Can you present a list of the postulates you have in mind?

 

[Popper]

The reason that the postulates seem to come out of nowhere is because we live in a macroscopic world and do have a built in physical intuition of quantum mechanics.

I'll post the postulates for further conversation referance (for simplicity assume non-degenerate discrete spectrum):

First postulate: At a fixed time t the state of a physical system is defined by specifying a ket |psi(t)> belonging to the state spce E

Second postulate: Every measurable physical quantity A is described by an operator A actin in E; this operator is an observable.

Third postulate: The only possible result of a measurement of a phyical quantity A is one of the eigenvalues of the corresponding observable A.

Fourth postulate: When the physical quantity A is measured on a system in the normalized state |psi>, the probability of obtaining the non-degenerate eivenvalue a_n of the corresponding observable A is:

p(a_n) = |<u_n|psi>|²

where |u_n> is the normalized eigenvector of A associated with the eigenvalue a_n,

Fifth postulate: If the measurement of the physical quantity A on the system in the state |psi> gives the result a_n, the state of the system immediately after the measurement is the normalized projection P_n|psi>/sqrt(<psi|P_n|psi>) of |psi> onto the eigen subspace associated with a_n.

Sixth postiulate: The time evolution of the state vector |psi(t)> is governed by the Schrodinger equation:

ihbar (d/dt)|ps> = H(t)|psi>

where H(t) is the observabe associated with the total energy of the system.

 

[ZapperZ]

I find the fundamental postulates of QM very ad-hoc and strange.
Compare them to the fundamental postulates of special relativity, special relativity naturally arises out of classical electromagnetism and the equivalence of all inertial frames, while QM seems to come out of nowhere.

It does?

The postulate of SR appears "ad hoc" to me as well. After all, one HAS to make such a thing when saying that c is constant in all reference frame. Only AFTER making a logical/mathematical consequence of such a postulate, and having it match to physics observation, can one consider such a postulate to be correct.

So I don't see this being any different than any other postulate made in physics.

Zz.

 

[tom.stoer]

I find the fundamental postulates of QM very ad-hoc and strange.
Compare them to the fundamental postulates of special relativity, special relativity naturally arises out of classical electromagnetism and the equivalence of all inertial frames, while QM seems to come out of nowhere.

Could it be that you are not bothered by the postulates but by the entities used in the postulates?

 

[HomogenousCow]

Could it be that you are not bothered by the postulates but by the entities used in the postulates?

More or less, my point is that there is a natural procession from classical mechanics to SR and even GR, the system of things are still comparable to each other, we still have our good old equations of motion and initial value problems.
Whereas for quantum mechanics, it seems the entire formalism has changed. Probability is involved, complex numbers and observables becoming operators, it is completely departed from all other theories.

 

[HomogenousCow]

It does?

The postulate of SR appears "ad hoc" to me as well. After all, one HAS to make such a thing when saying that c is constant in all reference frame. Only AFTER making a logical/mathematical consequence of such a postulate, and having it match to physics observation, can one consider such a postulate to be correct.

So I don't see this being any different than any other postulate made in physics.

Zz.

I disagree, Einstein's postulates seem much more physical and "tangible" and so do Newtons.
The QM postulates just aren't really motivated by experiment and they don't seem to originate from some simple physical principle.

 

[ZapperZ]

I disagree, Einstein's postulates seem much more physical and "tangible" and so do Newtons.
The QM postulates just aren't really motivated by experiment and they don't seem to originate from some simple physical principle.

I disagree. SR postulate did not originate from "some simple physical principle" either! What physical principle dictates the uniformity and the isotropic nature of c?

What you are describing appears to be a matter of tastes, and that is what we are arguing about now.

Zz.

 

[BruceW]

@ HomogenousCow: I know what you mean, in that the QM theory seems wildly different to any other physical theory. But I would say that the QM postulates are motivated by experiment. I think they are just the simplest set of postulates that people could think of that matched the new idea of the quantum. (Edit: and this idea of the quantum seemed to consistently agree with experiment).

QM is the same as other physical theories in that it has postulates, makes predictions. But it seems very different to most other of the broad physical theories. You could say this is depends on the person. Some people might think that QM is similar to other theories, and other people might think QM is very dissimilar. Neither person is necessarily wrong. All we can say is that so far QM has not been proven wrong. Therefore, it obeys the correspondence principle, so its result agrees with other theories (like classical mechanics), in the limit which those theories apply to.

 

[BruceW]

What you are describing appears to be a matter of tastes, and that is what we are arguing about now.

yeah, HomogenousCow is not disputing that QM is a physical theory. he's just saying that it looks pretty ad-hoc and strange. And I would agree with him. Maybe this is too 'philosophical' for PF.

 

[vanhees71]

There is nothing too philosophical with quantum mechanics. It's a clearly stated physical theory about the behavior of matter, it's giving a very precise description about what's observed in nature, and so far there is no empirical evidence for a restriction of its validity. That's why it is the most fundamental theory about matter (its constituents, the elementary particles and the interactions between them). So there is no doubt that quantum theory is a very nice example for a physical theory.

Surely, quantum theory leads to a quite drastic correction of our everyday experience about the behavior of matter, but that's only due to the fact that we are simply not confronted with situations, where the quantum effects (coherence) play a role. Of course, given the microscopical structure of matter the most important conclusion from quantum mechanics is the supposedly "simple fact" that matter in everyday life is pretty stable. Given this "simple fact" in almost all circumstances of everyday life classical mechanics and classical electromagnetism "explain" the behavior of objects around us pretty well.

It's, of course, important for the consistency of these different layers of description of nature that all of them can be understood in terms of the most fundamental theory at hand, which is, in our context quantum theory. That's indeed the case, because the classical behavior of macroscopic objects, including the absence of the unusual properties of coherence, entanglement, "wave-particle complementarity", etc. is understandable from quantum many-body theory, where one derives the classical equations for macroscopic objects as a very good approximation to the full quantum behavior, which we cannot resolve with our senses anyway.

This is not so different from the theory of relativity, where also many aspects appear not to be so easily comprehensible with everyday experience. This starts even with the most simple kinematical effects like the relativity of simultaneity, time dilation, length contraction, etc. Again, the reason is that we are not used to circumstances where relativistic effects play a prominent role since usually bodies around us do not move with a speed close to the speed of light and we are living only in a very weak gravitational field of the Earth (and other bodies around us like the moon and the sun). So again, Newtonian mechanics is well valid in our world of everyday experience, and that's why we are unused to the fact that these phenomena are descibable by the Newtonian approximation of the more comprehensive theory of relativity.

All that is of course not "philosophical" but at the very heart of the natural sciences and the role of model building within them.

 

[HomogenousCow]

I disagree. SR postulate did not originate from "some simple physical principle" either! What physical principle dictates the uniformity and the isotropic nature of c?

What you are describing appears to be a matter of tastes, and that is what we are arguing about now.

Zz.

Right but I find the postulate that speed of light is the same in all inertial frames much more physical and "tangible" than the QM postulates.
If you look at it, QM is very much just infinite dimensional linear algebra given physical interpretations, it baffles me why they correctly describe our world.

The whole business of quantization gives me a weird feeling, of them all canonical quantization is the most unintuitive to me and path integral quantization is least.

 

[HomogenousCow]

But I would say that the QM postulates are motivated by experiment. I think they are just the simplest set of postulates that people could think of that matched the new idea of the quantum.

I agree but it baffles me why this model works, none of the postulates are directly motivated by experimental evidence, only after some deep digging do we find that they agree with interference and other observations.

 

[Jorriss]

I agree but it baffles me why this model works, none of the postulates are directly motivated by experimental evidence, only after some deep digging do we find that they agree with interference and other observations.

The postulates are a refinement of years of progress based on experimental work. Quantum mechanics wasn't birthed by writing down the postulates.

 

[WannabeNewton]

It seems like your uneasiness with the postulates of QM is that they are mathematical and not "physical" like the postulates of SR? While I agree that the motivation for the postulates of SR from the need to make Maxwell's equations frame invariant is extremely "physical" and elegant, it is not as if the postulates of QM are completely arbitrary even if they are mathematical. Read the first couple of pages in ch2 of Ballentine and see if the motivation there is satisfactory enough.

 

[micromass]

I find the QM axioms rather natural if you take the approach of C*-algebra. In that approach, you can clearly see the link between QM and classical mechanics. And you clearly see that QM and CM are the same thing, except that QM is noncommutative. The usual axioms are than derived through some difficult mathematics.

 

[Fredrik]

Edit: LOL. micromass always finds a way to say something similar to what I'm going to say while I'm typing. (Although, it's been a while since the last time that happened).

There are at least two other approaches to QM (other than starting with the Hilbert space axioms) that are a bit more intuitive. Unfortunately, they require some very heavy math. I don't have a perfect understanding of either of these approaches, so it's possible that some of these details are a bit off, but I'll try:

One approach argues that if a theory can predict the average value of a sequence of measurements on identically prepared systems, done by identical measuring devices, then we should be able to define equivalence classes of measuring devices, and some mathematical operations on the set of equivalence classes that give it the structure of a C*-algebra. The rest is just an application of the theory of representations of C*-algebras. In particular, there's a theorem that ensures that there's a homomorphism from the C*-algebra (whose members are called "observables" btw) into the set of bounded operators on a Hilbert space. This approach is called algebraic quantum mechanics.

Apparently if the C*-algebra is commutative, there's some other theorem that ensures that what we get is a classical theory.

Another approach argues that there should be a lattice (a partially ordered set that satisfies some additional conditions) associated with each theory, and that this lattice will need to satisfy some technical conditions in order to not be extremely hard to work with. Apparently these technical conditions are sufficient to ensure that the lattice is isomorphic to the lattice of Hilbert subspaces of a Hilbert space. This approach is called quantum logic.

Apparently, if the lattice satisfies some additional conditions that makes it even easier to work with, it will be isomorphic to the (partially ordered) set of subsets of some set X. This set can then be interpreted as the phase space of a classical theory.

Some not so easy references:

Strocchi: Introduction to the mathematical structure of quantum mechanics: a short course for mathematicians.
Araki: Mathematical theory of quantum fields.
Varadarajan: Geometry of quantum theory.

For the algebraic approach, you need to know lots of topology and functional analysis. The book by Varadarjan is so hard to read that it's even hard to tell what sort of things you need to know for the quantum logic approach. The book contains long sections on projective geometry, and measure theory on locally compact simply connected topological groups.

 

[BruceW]

I agree but it baffles me why this model works, none of the postulates are directly motivated by experimental evidence, only after some deep digging do we find that they agree with interference and other observations.

yeah, true. But if you look at the postulates of classical mechanics, then it is not immediately obvious that they are related to experimental evidence. As far as I know, these are the most commonly used set of postulates:

1) Transformations between inertial frames of reference are described by continuous, differentiable and bijective functions.
2)If the velocities of two freely moving particles are equal in system S, they will also be equal in system S'.
3)All the inertial reference frames are equivalent.
4)The space in any inertial reference system is isotropic.
5)Simultaneity is not relative. ((This is the one that is different for special relativity))

Maybe you would count these as postulates about the geometry we exist in, and stuff like F=ma as the postulates of mechanics... But the point is that you need these 'geometry postulates' as well as the 'F=ma' postulates, to get classical mechanics. You can't just start with F=ma. In a similar way, you need to have some postulates in QM which do not give off an obvious appearance of being motivated by experimental evidence.

So I'd say that (in my personal opinion), the postulates of QM give as much of an appearance of 'being motivated by experimental evidence' as the postulates of classical mechanics are. (And to be clear, I know it is true a priori that both sets of postulates are motivated by experimental evidence. But the only thing I am discussing is whether it is immediately obvious from looking at the postulates that they are motivated by experimental evidence).

 

[WannabeNewton]

You might be interested in this document: http://www.math.uchicago.edu/~may/VIGRE/VIGRE2009/REUPapers/Gleason.pdf

Jorriss (and some others at Uchicago to be fair xD) showed it to me ages ago but I guess he forgot to link it himself; esentially it outlines the same issues that you are putting forth but then goes on to give the more "classical mechanics related" C* algebraic approach (note that the results aren't proven in this document but there are references to texts e.g. Rudin where you can see the associated proofs).

 

[micromass]

Fredrik, you mention two different theories. Can you tell me which theories are described in the books you recommend?

 

[Fredrik]

Fredrik, you mention two different theories. Can you tell me which theories are described in the books you recommend?

Strocchi and Araki cover the algebraic approach. Varadarajan covers the quantum logic approach. If you are interested in the quantum logic stuff, it might be easier to start with Beltrametti & Cassinelli, or Piron. (Not sure though, I have only read small parts of each).

I should also add that I have only read the first few pages of Araki. They are very readable by the way. There he talks about defining equivalence classes of measuring devices and that kind of stuff. Since the goal of the book apparently has something to do with quantum fields, Strocchi is probably a much better choice for the basics of the algebraic approach.

 

[Jano L.]

I think some of the reasons why quantum theory looks so strange in comparison to other theories are:

- the presentation one encounters in schools/books involves rather contradictory notions (wave - particle). Bohr tried to make this an advantage by his complementarity principle, but after years it still makes a lot of people confused;

- It is very mathematical; Jaynes once remarked that the quantum theory is a mathematical shell without sensible physical content. It is indeed a fusion of new pieces of mathematics due to many different people with different physical/philosophical thinking - Schroedinger, Heisenberg, Dirac, von Neumann ... The mathematics was new and difficult at the time and I think this mystified its physical content very much;

- it still has basic important questions unanswered, like how the actual events that happen are to be described or what is the link to the common macroscopic physics.

 

[WannabeNewton]

The problem is that people learn classical mechanics using the usual "physical" approach but never learn classical mechanics (CM) in a very mathematical manner (unless they chose to on their own of course) thus giving rise to the issues that the OP puts forth. You can formulate CM in an algebraic manner as well and then go on to QM and the discomfort wouldn't be as big in magnitude. It's like if you first learned about Lagrangian and Hamiltonian mechanics using the language of smooth manifolds and then went on to GR, things would feel very familiar. Anyways, in the spirit of the above link, see here for a brief summary: http://www.math.uchicago.edu/~may/VIGRE/VIGRE2011/REUPapers/Gleason.pdf

 

[BruceW]

probably a large part of the reason why I find the QM postulates ad-hoc and strange is because they talk about the system, and the state of the system, and how this is related to observables. So straight away, it looks (to me) like the theory of QM is like "well, we've got these strange experimental results, so let's just put together a theory that links the system with the observations, physical meaning be damned"

So really, for me, the QM postulates are strange because when I read them, I think "the people who wrote these really didn't have faith in what they were writing". Of course, this is just a matter of tastes. This doesn't discredit QM in any way as a physical theory. And people have put their faith in QM. For example, Paul Dirac, who predicted the positron, saw that the QM equations might suggest that there is a positively charged electron, and so he predicted the positron, by trusting in the equations of QM.

p.s. when I say 'faith', don't take this as a 'belief in a deity' kind of faith. You know what I mean really.

 

[Fredrik]

probably a large part of the reason why I find the QM postulates ad-hoc and strange is because they talk about the system, and the state of the system, and how this is related to observables. So straight away, it looks (to me) like the theory of QM is like "well, we've got these strange experimental results, so let's just put together a theory that links the system with the observations, physical meaning be damned"

It looks that way to me too. To be more specific, it looks like a toy theory that was created by a mathematician just to show it's possible to define a theory that assigns non-trivial probabilities (i.e. not always 0 or 1) to possible results of experiments. In fact, it looks like the simplest such theory that has any chance of being even a little bit useful. So I find it pretty remarkable that it has turned out to be the best theory science has ever produced.

 

[BruceW]

The problem is that people learn classical mechanics using the usual "physical" approach but never learn classical mechanics (CM) in a very mathematical manner (unless they chose to on their own of course) thus giving rise to the issues that the OP puts forth. You can formulate CM in an algebraic manner as well and then go on to QM and the discomfort wouldn't be as big in magnitude.

that's a good point. people don't usually begin with a mathematical description of CM. And the link you gave talks a lot about the state, for a classical system. So it kind of answers my last post, by saying that actually CM involves a lot of talk about 'the state'.

Although I think that this stuff about 'the state' is not in every mathematical formalism of CM? As in, I would guess it is possible to do a mathematical formalism of CM without defining states as "linear functionals on the set of observables" and without similar statements like these? I am not sure, I really have no experience in this area.

But then on the other hand, if QM and CM are similar in this mathematical formalism (using states and observables), then this means that QM and CM are similar generally. (Since 'similarity' between two physical phenomena should not be dependent on the mathematical formalism). i.e. I am assuming any mathematical formalism of CM that does not involve 'states and observables' should be equivalent to the mathematical formalism of CM that does involve 'states and observables'. (equivalent in the sense that it describes the same physical phenomena).

 

[micromass]

that's a good point. people don't usually begin with a mathematical description of CM. And the link you gave talks a lot about the state, for a classical system. So it kind of answers my last post, by saying that actually CM involves a lot of talk about 'the state'.

Although I think that this stuff about 'the state' is not in every mathematical formalism of CM? As in, I would guess it is possible to do a mathematical formalism of CM without defining states as "linear functionals on the set of observables" and without similar statements like these? I am not sure, I really have no experience in this area.

But then on the other hand, if QM and CM are similar in this mathematical formalism (using states and observables), then this means that QM and CM are similar generally. (Since 'similarity' between two physical phenomena should not be dependent on the mathematical formalism). i.e. I am assuming any mathematical formalism of CM that does not involve 'states and observables' should be equivalent to the mathematical formalism of CM that does involve 'states and observables'. (equivalent in the sense that it describes the same physical phenomena).

States in CM are usually described as actual points in some space. So you have a space $X$ (for example $\mathbb{R}^{6n}$) and the points in here are states. The observables are continuous functions $f:X\rightarrow \mathbb{R}$. So the set of all continuous functions $\mathcal{C}(X)$ is the set of observables, in some sense.

In the "algebraic" approach to QM, you are given a set of observables $A$ (which is analgous to $\mathcal{C}(X)$). But the states now are certain linear functionals on $A$. So a state is a function $T:A\rightarrow \mathbb{R}$ such that some properties are preserved. If $a\in A$, then we should see $T(a)$ as some kind of average.

These two approaches seem not very similar, but they actually are. Indeed, given a classical system $X$ with $\mathcal{C}(X)$, then any $x\in X$ determines a linear functional as $\mathcal{C}(X)\rightarrow \mathbb{R}: f\rightarrow f(x)$. Furthermore, it can be shown that this correspondance is a bijection. So all linear functionals (with the right properties) correspond to elements of the state space.

This correspondance fails if $A$ is noncommutative. And if $A$ is noncommutative, then there is no right notion of state space (there are notions, but they are very ill behaved). So instead of working with an actual space $X$, we work with the linear functionals as states. Then QM can be developed by associating a Hilbert space with $A$ and so on.

 

[HomogenousCow]

The problem is that people learn classical mechanics using the usual "physical" approach but never learn classical mechanics (CM) in a very mathematical manner (unless they chose to on their own of course) thus giving rise to the issues that the OP puts forth. You can formulate CM in an algebraic manner as well and then go on to QM and the discomfort wouldn't be as big in magnitude. It's like if you first learned about Lagrangian and Hamiltonian mechanics using the language of smooth manifolds and then went on to GR, things would feel very familiar. Anyways, in the spirit of the above link, see here for a brief summary: http://www.math.uchicago.edu/~may/VIGRE/VIGRE2011/REUPapers/Gleason.pdf

I think there would still be some discomfort, I'm guessing the born interpretation doesn't just pop out of this reformulation of QM.
The fact that the theory is ultimately undeterministic, I think, is what ultimately makes me uncomfortable.
To expand, let's say that the hidden variables theory is correct and we one day find a deterministic theory that explains it all, how does one explain the fact that we just happened to have created an indeterminstic one beforehand?? How did we happen to guess that these were the statistical version of the underlying deterministic theory?

 

[vanhees71]

Well, you can think about determinism as something much more incomprehensible than indeterminism. Why should nature behave deterministic?

Then, of course there is always the possibility that one day any physical theory may be found to be incomplete. It's even pretty likely that this happens to a physical theory. Quantum theory is not an exception. Only at the moment, there is no hint, how such a theory might look like. However, if such a theory is deterministic, it must be a pretty complicated non-local scheme since the violation of Bell's inquality (in accordance with quantum theory!) is a well-established empirical fact. So far there is no convincing non-local theory found. As long as we have nothing better we have to live with quantum theory. Due to the great successes of quantum theory it is clear that any more comprehensive theory must contain it as a limiting case of approximate accuracy and with a certain range of applicability. This is the case with Newtonian mechanics vs. the theory of relativity. The former can be derived from the latter as an approximation, valid for velocities small compared to the speed of light and for only small gravitational fields.

 

[Fredrik]

I think there would still be some discomfort, I'm guessing the born interpretation doesn't just pop out of this reformulation of QM.

It doesn't imply that we should assign probabilities to measurement results, but it turns out that there's only way to do that. That result implies the Born rule.

 

[micromass]

I think there would still be some discomfort, I'm guessing the born interpretation doesn't just pop out of this reformulation of QM.

In Classical mechanics, the observables can be seen as $\mathcal{C}(X)$ and the (pure) states are certain functionals. The Riesz representation theorem suggests that the states are averages with respect to some probability theory.

In the algebraic approach, we have an algebra of observables $A$ and certain functionals, called the (pure) states. If we only accept that there functionals are averages wrt to some probabilities (which is already suggested in the classical sense), then everything else pops out. In particular, the Born rule can be deduced. A good account of this is given in Ballentine. Ballentine sadly does not do the most general case of C*-algebras. But he does show how you can derive the entire probabilistic structure from simply accepting that the states give some kind of averages.

 

[micromass]

Apparently, in QM we need both pure states and other states. But in Classical Mechanics, we can get by with just the pure states. The pure states already determine the entire phase space $X$. The other states induce naturally probability distributions on $X$, but they don't seem necessary. Is there some interpretation in CM for states that are not pure states? Are they somehow needed in CM?

 

[HomogenousCow]

So to summarize, in this formulation of CM and QM, we have indeterminism everywhere but observables in CM all commute thus we have simultaneous eigenstates for all observables and the probabilities never come to play at all?

 

[Fredrik]

So to summarize, in this formulation of CM and QM, we have indeterminism everywhere but observables in CM all commute thus we have simultaneous eigenstates for all observables and the probabilities never come to play at all?

To view CM and QM as two aspects of the same thing, we need to view CM as an assignment of probabilities as well. But the probabilities assigned will all be 0 or 1, when we're dealing with pure states.

 

[HomogenousCow]

so just as I said then, does the schroedinger equation also arise from this?

 

[Charles Wilson]

From Age of Entanglement, Louisa Gilder, ISBN978-1-4000-9526-1:

"Thus started the central debate of the conference, in which, as Mermin remembered, "[[John]] Bell claimed that in some deep way quantum mechanics lacked the naturalness that all classical theories possessed." There was no problem with the interpretation of classical physics. For example, as [[Kurt]] Gottfried granted Bell, "Einstein's equations tell you" their own interpretation. "You do not need him whispering in your ear." In quantum mechanics, on the other hand, Gottfried admitted, even the greatest classical physicist "would need help: 'Oh, I forgot to tell you that according to Rabbi Born, a great thinker in the yeshiva that flourished in Gottingen in the early part of the 20th century, [the amplitude-squared of the Schrodinger equation] is' " to be interpreted as a probability. despite all indications to the contrary. Bell felt it was obvious that something profound was missing from quantum mechanics; "Kurt [[Gottfried]]", Mermin said, "never felt this in his bones."

FWIW, I see two confusions: Bohr is the author of one. He attempted to force an interpretation - a philosophical orientation - onto the scientific community and beyond. We are still paying that price. Bohr lays down an epistemological Kantianism that divides "das Noumena" from "Phenomena" by stating that you can never get to "das Noumena" - the quantum particle - without the Phenomena of the equipment AND our language cannot never get to a scientific treatment of "What is really there" because language cannot in principle be used to describe "What goes on".

Hegelianism awaits at this point: "If the scientist cannot get to "das Noumena", then there does not appear to be a "Das Noumena"." Science will consist of arguments over sentences describing classical world objects - ONLY!

The second confusion is more important. It is known as "Born's Interpretation" of the Schrodinger Equation for a reason. Schrodinger himself appeared at times to be at a loss for what his own equation meant. I again quote from the John Clauser interview I gave in another post: "We have no idea how we got from Schrodinger's waves to Born's dots on the screen."

If we squeeze all "non-positive" space out of our analyses, do we lose something? I think so:
Consider a "Toy Universe" where a "Positive Vacuum Value X^Y" is matched by a "Negative Vacuum Value X^ -Y". We might begin an analysis with a statement: "X^Y times X^ -Y = 1". OK. Fine. All upfront.

Suppose, without knowing it, we made use of a mathematical definition that took the absolute value of an exponential, we would have "X^Y times x^ |-Y| = X^ 2Y". We might eventually get to a statement such as, "This implies that the magnitude of the cosmological constant must be smaller than 1/(10^23 kilometers)^2. Our theoretical estimate suggesting a magnitude greater than 1/(1kilometer)^2 is incorrect by, at the very least, an astonishing factor of 10^46." Larry Abbott, Mystery of the Cosmological Constant, SciAm, May 1988.

If Guth's Cosmic Inflation is true, does it work in reverse? Could there be a Guthian "Big Crunch"? It would mean that at the last moments of the previous universe, SpaceTime collapsed to a volume to Unification by a factor of 10^ ~50ish or something. Did matter follow?

Philosophy has taken a few hits on this site recently and I think the response should be, "Not Guilty!" Bohr is overbearing at times, a person you would eventually steer away from at a party. Born was doing great things with the tools that were available to him. At the same time, others were chafing at making "an interpretive use" into the "only way to see it".

We still are with this today. I defer to ZapperZ and Dr. Chinese who have an understanding greater than I. THEIR use of language and math has not been hampered by The Restricted Categories of the Understanding.

Good for them!

CW

 

[Fredrik]

so just as I said then

Yes, close enough.

does the schroedinger equation also arise from this?

It does if we include the requirement that the C*-algebras we consider must have a subalgebra that corresponds to translations in time. (Or something like that; I don't know how to make the statement exactly right).

 

[BruceW]

Apparently, in QM we need both pure states and other states. But in Classical Mechanics, we can get by with just the pure states. The pure states already determine the entire phase space $X$. The other states induce naturally probability distributions on $X$, but they don't seem necessary. Is there some interpretation in CM for states that are not pure states? Are they somehow needed in CM?

I really don't get what you mean here. in QM we don't need mixed ensembles. We use them for the same reason we use mixed ensembles in CM. The only special thing about CM is that a pure ensemble is pretty boring, it will always give you the same measurement value, while a pure QM ensemble will generally give a different value each time.

edit: I probably didn't explain myself enough. and I know it is annoying when people do that. In QM, I would think that we don't need mixed ensembles for the same reason we don't need mixed ensembles in CM. In both cases, the idea of a mixed ensemble doesn't contain any extra physics (any more than the pure states involved). We are just assigning probability (in the conventional sense) to certain states. You can interpret this as our lack of information, or in the frequentist philosophy that it gives the fraction of systems out of a large bunch of systems, in the limit of a very large number of systems.

 

[Fredrik]

Apparently, in QM we need both pure states and other states. But in Classical Mechanics, we can get by with just the pure states. The pure states already determine the entire phase space $X$. The other states induce naturally probability distributions on $X$, but they don't seem necessary. Is there some interpretation in CM for states that are not pure states? Are they somehow needed in CM?

I'm not sure if they're needed, but it's not hard to think of a situation where a mixed state can be used. Consider two arbitrary pure states $s_1,s_2$. Flip a coin to decide whether to prepare the system in state $s_1$ or state $s_2$, and then tell your experimentalist friend that you did that, but don't tell him the result of the flip.

The experimentalist would now be correct to think of the system as being in a mixed state. Of course, he doesn't have to think about it in those terms. He can just think about it in terms of pure states and basic probability theory. So I don't think I can say that the concept of mixed states is really needed here. But maybe there are better examples.

 

[dextercioby]

I find the fundamental postulates of QM very ad-hoc and strange.
Compare them to the fundamental postulates of special relativity, special relativity naturally arises out of classical electromagnetism and the equivalence of all inertial frames, while QM seems to come out of nowhere.

Everything in life, including science, is a matter of subjectivity. You like that, you don't like that, what's not intuitive to you is for someone else. The more subjective you are, the less you know about a certain topic. GR and Quantum Mechanics are both equally valid generalizations of classical mechanics. Formulate classical mechanics in such a way that both GR and QM are the natural extensions of it. If you think that CM is F=ma + forces add together like vectors and the force of 1 on 2 is the opposite of the force of 2 on 1, then you're not ready for GR, not ready for QM and seeing the (diluted or not) axiomatization of QM would make you say: <This is weird. Where did it come from?>.

Advice: read more and don't be afraid of mathematics. The more math you know, the more logical will the advanced physics look to you.

 

[stevendaryl]

Advice: read more and don't be afraid of mathematics. The more math you know, the more logical will the advanced physics look to you.

I don't think that's completely true. It's partly a personality difference, but there are physicists who know quite a bit of the mathematics behind quantum mechanics who still think that it's strange and a bit mysterious. Feynman famously said "I think I can safely say that nobody understands quantum mechanics", and I don't think that can be attributed to his unfamiliarity with the advanced mathematics involved.

There is very difficult mathematics involved in General Relativity, too, but I personally don't find it very mysterious, in spite of its difficulty. It's not the difficult mathematics.

Some people have also suggested that it's the intrinsic nondeterminism in quantum mechanics that's bothersome. I don't think that's true, either. It's true that classical mechanics is deterministic, but I don't think that there's anything conceptually difficult about assuming the existence of random processes--perfect coin flips--whose outcomes are not determined.

No, I think that what strikes some people as weird about quantum mechanics is that it gives such a prominent role to the concept of an "observable". The C* algebras in this regard succeed in making classical mechanics sound as weird as quantum mechanics, rather than making quantum mechanics as intuitive as classical mechanics.

Now, of course science is concerned with observation, but I don't think that's the same thing as being about observables. The weird thing about making an observable a separate kind of object in quantum mechanics (or classical mechanics in the C* algebra approach) is that observers are themselves physical systems, and observations are themselves physical interactions between systems. In my opinion, a satisfying formulation of quantum mechanics would not postulate the existence of observables. It would describe how physical systems interact, and the properties of measuring devices would be special cases derivable from the general case.

 

[BruceW]

If you think that CM is F=ma + forces add together like vectors and the force of 1 on 2 is the opposite of the force of 2 on 1, then you're not ready for GR...

you mean it is important to learn stuff like Hamiltonian vector field, canonical transformation, action integral, that kind of stuff?

 

[stevendaryl]

Another related point. In the C* algebra formalism, the difference between quantum mechanics and classical mechanics is noncommutativity of observables. Now, classical mechanics has its share of noncommutativity; the matrices used to describe rotations and boosts are noncommutative. Noncommutativity is to expected for nontrivial operators. But the weird thing about quantum mechanics is the association between operators and observables. Why should an observation have anything to do with operators? Once again, if an observation is just a physical interaction between one system (a scientist, or one of his measuring devices) and another system (an electron, say), why isn't it just described by the evolution of the two systems? Why do operators come into play?

 

[HomogenousCow]

you mean it is important to learn stuff like Hamiltonian vector field, canonical transformation, action integral, that kind of stuff?

Learning these things before learning QM makes the mathematics more familiar but the physical situation is still very strange.

 

[HomogenousCow]

I don't think that's completely true. It's partly a personality difference, but there are physicists who know quite a bit of the mathematics behind quantum mechanics who still think that it's strange and a bit mysterious. Feynman famously said "I think I can safely say that nobody understands quantum mechanics", and I don't think that can be attributed to his unfamiliarity with the advanced mathematics involved.

There is very difficult mathematics involved in General Relativity, too, but I personally don't find it very mysterious, in spite of its difficulty. It's not the difficult mathematics.

Some people have also suggested that it's the intrinsic nondeterminism in quantum mechanics that's bothersome. I don't think that's true, either. It's true that classical mechanics is deterministic, but I don't think that there's anything conceptually difficult about assuming the existence of random processes--perfect coin flips--whose outcomes are not determined.

No, I think that what strikes some people as weird about quantum mechanics is that it gives such a prominent role to the concept of an "observable". The C* algebras in this regard succeed in making classical mechanics sound as weird as quantum mechanics, rather than making quantum mechanics as intuitive as classical mechanics.

Now, of course science is concerned with observation, but I don't think that's the same thing as being about observables. The weird thing about making an observable a separate kind of object in quantum mechanics (or classical mechanics in the C* algebra approach) is that observers are themselves physical systems, and observations are themselves physical interactions between systems. In my opinion, a satisfying formulation of quantum mechanics would not postulate the existence of observables. It would describe how physical systems interact, and the properties of measuring devices would be special cases derivable from the general case.

I agree with this.
The fact that the universe is nondeterministic at the basic level is something I can accept, but the way this is implemented in QM strikes me as odd and bothersome.
I think the sentence that captures my problem with the whole thing is, where does the indeterminism stop? Observers in QM seem to have to be classical and macroscopic in order for everything to make sense, the probabilities have to stop somewhere or else the whole concept of a frame of reference seems to be ill-defined.
It seems that the more I think about it, the more QM seems like an effective theory which acts as a bridge between the classical world and the quantum world.

 

[Fredrik]

But the weird thing about quantum mechanics is the association between operators and observables. Why should an observation have anything to do with operators? Once again, if an observation is just a physical interaction between one system (a scientist, or one of his measuring devices) and another system (an electron, say), why isn't it just described by the evolution of the two systems? Why do operators come into play?

I can't answer the part about why it's not just "evolution of the two systems", other than by saying that what these approaches have in common is that they can be justified by arguments based on the idea that a theory of physics assigns probabilities to measurement results. They're not based on requirements about how the theory is supposed to be a description of what's happening, or anything like that. They're just based on the idea of falsifiability.

Note also that the algebraic approach doesn't assume that observables are operators. They are introduced as equivalence classes of measuring devices, and then you define some algebraic operations on the set of observables that turn it into a C*-algebra. Some of them are very natural. For example, for each real number r and each observable A, there should be an observable rA that corresponds to a measuring device like this: Take any measuring device from the equivalence class A, and add a component that multiplies the result by r (so that the result of the measurement will be r times what it would be without the modification). Let rA be the equivalence class that contains the modified device.

Some of the other operations are probably just technical assumptions meant to give us something we can work with at all. But my point is that the observables are not defined as operators. The association with operators comes from a theorem about *-homomorphisms from C*-algebras into the set of bounded linear operators on a Hilbert space.

I think that even if we don't use the algebraic approach, there's a case to be made for why observables (equivalence classes of measuring devices) should be represented by operators. Unfortunately it's not clear enough in my head that I can put it together right now. I think that I would try to argue that observables should correspond to projection-valued measures, and then refer to the spectral theorem to associate them with self-adjoint operators. But I'm definitely in over my head here, so don't take this too seriously.

 

[stevendaryl]

I can't answer the part about why it's not just "evolution of the two systems", other than by saying that what these approaches have in common is that they can be justified by arguments based on the idea that a theory of physics assigns probabilities to measurement results.

It seems a little weird to me that is what a theory of physics should be about, because "measurement" is not fundamentally different from any other kind of interaction. We just interpret certain interactions as being "measurements" when they result in a strong correlation between a persistent record (photograph, bits on a hard drive, marks on paper, etc.) and some aspect of the universe that we are interested in measuring. The fact that it is a measurement is the significance that WE put on the interaction, but it seems weird to me to think that physics cares whether something is a measurement, or not.

Note also that the algebraic approach doesn't assume that observables are operators. They are introduced as equivalence classes of measuring devices, and then you define some algebraic operations on the set of observables that turn it into a C*-algebra.

Well, if you don't think of them as operators, then it seems to me that turning them into a C* algebra is a strange thing to want to do. We have algebraic operations on observables such as:
If f is an observable and g is an observable, then f g is an observable. But what does it mean to multiply observables? It makes sense to multiply the results of two operations (interpreted as giving real numbers), but what is the meaning of multiplying the observables?

Some of them are very natural. For example, for each real number r and each observable A, there should be an observable rA that corresponds to a measuring device like this: Take any measuring device from the equivalence class A, and add a component that multiplies the result by r (so that the result of the measurement will be r times what it would be without the modification). Let rA be the equivalence class that contains the modified device.

Some of the other operations are probably just technical assumptions meant to give us something we can work with at all. But my point is that the observables are not defined as operators. The association with operators comes from a theorem about *-homomorphisms from C*-algebras into the set of bounded linear operators on a Hilbert space.

Well, I think the ones that are hard to interpret is multiplication of two observables. What does it mean? People informally say that A B means "first measure B", then measure A", but that's a little strange to interpret that as multiplication, because those two measurements take place at slightly different times.

I think that even if we don't use the algebraic approach, there's a case to be made for why observables (equivalence classes of measuring devices) should be represented by operators.

What makes a device a "measuring device"? It seems to me that it is the theory itself that tells us that under certain circumstances a microscopic fact (that an electron has spin-up along a certain axis) results in a persistent macroscopic fact (that a dot on a photographic plate appears on the left-hand side, rather than the right-hand side, or whatever).

Also, what is the notion of "equivalence" here? Two devices are equivalent if they ... what? Measure the same observable? That's a little circular, but what notion of equivalence are we supposed to be using?

Unfortunately it's not clear enough in my head that I can put it together right now. I think that I would try to argue that observables should correspond to projection-valued measures, and then refer to the spectral theorem to associate them with self-adjoint operators. But I'm definitely in over my head here, so don't take this too seriously.

 

[stevendaryl]

They [observables] are introduced as equivalence classes of measuring devices, and then you define some algebraic operations on the set of observables that turn it into a C*-algebra.

In my opinion, calling an observable of a C*-algebra an "equivalence class of measuring devices" is more suggestive than rigorous. I would think that if one really wanted to seriously talk about equivalence classes, then one would have to

1. Define what a "measuring device" is.
2. Define an equivalence relation on measuring devices.
3. Define the operations on measuring devices (addition, multiplication, scaling, or whatever).
4. Prove that the equivalence relation is a congruence with respect to those operations.

I don't think you can really do that in a noncircular way, because to make sense of the claim that a particular device is a measuring device for the z-component of spin angular momentum of some particle, you would need to assume some kind of dynamics whereby the device interacts with the particle so that its state evolves to a persistent record of the z-component of the spin angular momentum. You need to have a theory of interactions before you can ever know that something is a measuring device. So it's a bit weird to put in equivalence classes of measuring devices at the beginning, as opposed to having them come out of the theory.

 

[Fredrik]

It seems a little weird to me that is what a theory of physics should be about,

A set of statements about the real world must be falsifiable in order to be considered a theory of physics, and to be falsifiable, it must (at least) assign probabilities to possible results of experiments. This appears to be the absolute minimum requirement. This is why all theories involve probability assignments to results of measurements. It's not that measurements are fundamentally different from other interactions. It's just that this is part of what we mean by the word "theory".

Well, if you don't think of them as operators, then it seems to me that turning them into a C* algebra is a strange thing to want to do. We have algebraic operations on observables such as:
If f is an observable and g is an observable, then f g is an observable. But what does it mean to multiply observables?

I checked my copy of Strocchi, and the argument is quite complicated and ends with a comment that it shouldn't be considered a proof that we must use C*-algebras.

It certainly looks like multiplication is by far the algebraic operation that's the hardest to justify. All the others are fairly easy to justify.

Also, what is the notion of "equivalence" here? Two devices are equivalent if they ... what? Measure the same observable? That's a little circular, but what notion of equivalence are we supposed to be using?

Something like this:

Let E(A|s) denote the theory's prediction for the average result of a long series of measurements using measuring device A on objects of the type that the theory is about (e.g. electrons) that have all been subjected to the same preparation procedure s just before the measurement.

Two measuring devices A and B are said to be equivalent if E(A|s)=E(B|s) for all preparation procedures s.

 

[stevendaryl]

A set of statements about the real world must be falsifiable in order to be considered a theory of physics, and to be falsifiable, it must (at least) assign probabilities to possible results of experiments. This appears to be the absolute minimum requirement. This is why all theories involve probability assignments to results of measurements. It's not that measurements are fundamentally different from other interactions. It's just that this is part of what we mean by the word "theory".

But the C*-algebra approach sure seems to single out measurements (or observables) as being something different. As I said, the fact that some interaction is a measurement of some quantity is not what you start with, it's a conclusion. There's a long chain of deductions involved in reaching that conclusion. It seems weird that measurements, which are complicated, macroscopic phenomena very indirectly connected with the microscopic phenomena being described by theory, should appear in the theory as the fundamental objects of interest. That just seems like a bizarre mismatch between the elegance and simplicity of the observables in C*-algebra and the complexity and messiness of actual measuring devices.

Clearly, there's abstraction and/or idealization going on, but what is the nature of this idealization?