Can you define quantum mechanics in just one sentence?

[Ken G]

Ken, is your statement correct? I think not:

1. Does a PA (being complex) sum to 1?

2. AFAIK, a PA is not a 'probability' in any sensible sense.

That's right, that's why I am talking about probabilities not amplitudes. This is indeed the whole point-- amplitudes appear in mundane classical wave mechanics, what is different about quantum mechanics is:
1) it deals in probabilities
2) they can be negative (or even complex, though Aaronson tries hard to be able to talk about the phenomenon without ever mentioning complex numbers at all because he views it as an unnecessary complication to the basic issue).
Did you read his lecture?

 

[PatrickPowers]

That's right, that's why I am talking about probabilities not amplitudes. This is indeed the whole point-- amplitudes appear in mundane classical wave mechanics, what is different about quantum mechanics is:
1) it deals in probabilities
2) they can be negative (or even complex, though Aaronson tries hard to be able to talk about the phenomenon without ever mentioning complex numbers at all because he views it as an unnecessary complication to the basic issue).
Did you read his lecture?

In my opinion complex numbers are an unnecessary complication, but what they are representing is essential. I like the approach Feynman took in QED, where as far as I recall he never mentions complex numbers but instead uses the simple geometrical idea of a clock with one hand. One may model amplitudes and their cancellations in this simple and intuitive geometric way.

 

[Gordon Watson]

That's right, that's why I am talking about probabilities not amplitudes. This is indeed the whole point-- amplitudes appear in mundane classical wave mechanics, what is different about quantum mechanics is:
1) it deals in probabilities
2) they can be negative (or even complex, though Aaronson tries hard to be able to talk about the phenomenon without ever mentioning complex numbers at all because he views it as an unnecessary complication to the basic issue).
Did you read his lecture?

Thanks Ken, but sorry: I don't get your point!

You say "That's right" -- which I take to mean that you agree that your statement is wrong -- but you then continue with another confusion, your 1) + 2) above: implying that PROBABILITIES can be negative?

In my view, QM deals with probabilities derived from probability amplitudes. Thus, for me, QM is an important extension of ordinary probability theory: Especially when you realize that any probability density may be represented as the absolute square of a complex Fourier polynomial.

Then, seeing no point in belaboring our differences, but to be clear about what I mean by a probability: The probability P(A|C) is the expected proportion of long-run experimental outcomes, under condition C, in which A occurs. It is an estimate of the relative frequency of A that will be revealed by measurements under C.

Since I never expect a negative proportion, and have never heard of one being measured, I'm confident that probability needs no confusing negativities.

And Yes; I read and re-read the lecture with interest and fondness and pleasure and delight: seeing it compatible with the view that I've just expressed here.

With best regards,

Gordon

 

[jambaugh]

I don't see how that answers the question. A mole of gas at STP (using the NIST standard) has a temperature of 293.15 K and a pressure of 101325 Pa. That is, 293.1500000... and 101325.00000... Where is the infinite information content?

Take an infinite signal, transcribe it to binary, use the string of bits to represent a binary number 0.bit bit bit... and you have a real value between 0 and 1. Given the classical assumption of continuous energy (or temperature, or pressure, or ...) it takes infinite information to represent an exact state. Of course pragmatically we measure only up to some level of precision and generally assume some bounds on values. It's the infinite number of classical states between two values of a continuous observable with which idealistically we can encode infinite information in a classical system.

On the other hand, consider a quantum system: an electron has a magnetic dipole moment. This is usually quantified in terms of a "g factor", which has a http://physics.nist.gov/cgi-bin/cuu/Value?gem|search_for=g+factor" g=2.002 319 304 361 53(53) for electrons. In principle this value can be measured to arbitrary precision, and of course there is no reason to think the true value is rational, or even an algebraic irrational. And yet we use the techniques of quantum field theory to approximate g. This would seem to show a quantum mechanical system with infinite information content.

I can express it with one symbol, g. The fact that it is a fixed constant precludes you using it to actually encode a signal. There's no information content to its value since it is not a variable.

It's not the continuity of the gauge group that makes the system quantum mechanical. We can study lattice gauge theory for the http://thy.phy.bnl.gov/~creutz/z2/z2.ps" , for example.

I'm not sure I see the quantum mechanics in this model. It describes a discrete computational model, not a theory of a physical system. If Z2 is a symmetry of some quantum system and doesn't act trivially then there's at least 2 modes in the Z2 orbit and by quantum theory there is thus at least an SU(2) symmetry incorporating all the superpositions of those two modes.

Mind you there are discrete symmetries e.g. CPT. But they occur as symmetries of the theory not of the system. Like Born duality (x <-->p).

I think the "quantum" in quantum mechanics is there for historical reasons. Not every quantum mechanical system has discrete states. I suppose you could argue that quantum field theory is "beyond" mere quantum mechanics, but certainly by the time you get to quantum field theory you have to deal with the infinite variety of contributions to even the most fundamental processes.
BBB

QFT is still within QM. The "2nd quantization" is rather quantification (going from one to many). Now while one may argue for infinite information due to the e.g. infinite bosonic fock space and infinite momentum spectrum of field quanta, you'll note these are sources of divergence in the theory. When the theory is regularized one typically has finite information content. E.g. a system of photons in a box with reasonable upper limits to energy (no BH formation e.g.) is a finite dimensional system. The infinities in QFT are there for pragmatic reasons within the calculus not as a necessary fundamental assumption. (And I'm working on a paper suggesting bosonic fields should be quasi-bosonic with a finite upper bound on particle number.)

But I did make one mistake in my attempt. I shouldn't have said "symmetry" I should have said "transformation group" or "relativity group". One can have a perfectly valid quantum system with absolutely no symmetries. But of course one always has at the very least the one parameter group of time translations generated by the Hamiltonian. I.e. the dynamics.

JB:DEF3=Quantum mechanics is the physics of systems with continuous transformation groups and finite information content.

 

[bhobba]

I rather like the two axioms found in Ballentine - QM - A Modern Approach

1. To each dynamical variable there is a Hermitian operator whose egienvalues are the possible values of the dynamical variable.

2. To each state there corresponds a unique state operator P that must be Hermitian, non-negative and of unit trace. The average <A> for a dynamical variable with operator A in the virtual ensemble of events that may result from the preparation procedure for that state is <A> = Tr(PA).

Of course that pins it to the ensemble interpretation which I am sure not everyone agrees with.

Stuff like Schrodinger's Equation is derived from Galilean Invariance.

Thanks
Bill

 

[bhobba]

In my view, QM deals with probabilities derived from probability amplitudes. Thus, for me, QM is an important extension of ordinary probability theory: Especially when you realize that any probability density may be represented as the absolute square of a complex Fourier polynomial.

Bingo - for me as well.

Consider a system with N possible outcomes each with probability Pi. Write them as a row vector and expand in bra-ket notation. u = P1 |b1> + ... + PN |bn>. This is all vectors with positive entries such that add up to one. A perfectly good norm but in physics you generally use the inner product norm so map it to sqrt(Pi) instead and you end up with all vectors in this basis that are not negative, of unit norm and with Pi = |<u|bi>|^2/<u|u>. First problem is that in physics no basis is special so if we shift to another basis legitimate vectors will have a different representation and we can tell what basis we are using. To get around that we need to map it to the entire vector space which is easily done since in the formula Pi can be calculated for any vector. Secondly notice it is invariant to the transformation cu where c is any complex number suggesting it should be a complex vector space. And lastly since no basis is special we have to assume any orthonormal basis is a possible list of outcomes of some observation which immediately leads to the superposition principle and for any vectors a and b with b of unit norm |<a|b>|^2/<a|a> gives the probability of observing a system in state a to see if state b is the result.

It can be developed further that way by assigning values ai to the outcomes and defining a linear transformation A by A|bi> = ai|bi>, and we have <A> = sum (aiPi) = sum (ai <u|bi> <bi|u>) = sum (<u|ai|bi><bi|u>) = sum (<u|A|bi><bi|u>) = <u|A|u>. But I think this is enough to get the idea across it is simply a novel extension of probabilities by encoding them in a vector space such that basis independence holds.

Thanks
Bill

 

[Ken G]

You say "That's right" -- which I take to mean that you agree that your statement is wrong -- but you then continue with another confusion, your 1) + 2) above: implying that PROBABILITIES can be negative?

The "that's right" was in response to your unarguably true statement that probablity amplitudes are different from probabilities. I then went on to say that both I and Aaronson are talking about probabilities, not probability amplitudes, although we are also talking about contributions to a net probability result. The amplitude comes prior to the 2-norm that Aaronson is talking about, but what is not interesting is that the amplitudes could be negative (which has a simple classical wave analog), what's interesting is that the probablity contributions can be negative (which has no classical wave analog). And not just negative like a correction for double-counting of correlated outcomes, but fundamentally negative like the way interference works-- except it is interference in the probability contributions, not just fluxlike contributions as in classical wave interference.

In my view, QM deals with probabilities derived from probability amplitudes. Thus, for me, QM is an important extension of ordinary probability theory: Especially when you realize that any probability density may be represented as the absolute square of a complex Fourier polynomial.

I believe that is just exactly what Aaronson is saying when he talks about negative probabilities-- they are the kind of probabilities you can get from probability amplitudes. That is saying something very important about quantum mechanics, it is one of the key innovations, that probabilities can come from complex probability amplitudes (and so can be negative). If you don't want to mention the complex amplitudes, you just do the whole thing at the level of the negative probabilities. It's just a streamlined way to say the same thing as what you are saying is key about quantum mechanics.

Then, seeing no point in belaboring our differences, but to be clear about what I mean by a probability: The probability P(A|C) is the expected proportion of long-run experimental outcomes, under condition C, in which A occurs. It is an estimate of the relative frequency of A that will be revealed by measurements under C.

Since I never expect a negative proportion, and have never heard of one being measured, I'm confident that probability needs no confusing negativities.

But you are missing that Aaronson is not talking about negative net probabilities, he is talking about negative probability contributions in any sum of ways that A can occur under condition C. So it is just exactly the kind of probability that you mean, he is merely pointing out the key innovation that we are going to allow the contributing terms to be negative. In other words, if A can occur under condition C in way X, and in way Y, then A might be less likely to occur than in way X by itself-- if way Y corresponds to a negative probability. That's all he's saying, that's the central crux. He is trying to make it simple, on purpose.

And Yes; I read and re-read the lecture with interest and fondness and pleasure and delight: seeing it compatible with the view that I've just expressed here.

No one ever said it was incompatible with the view you just expressed, indeed that is the whole point. Probabilities that come from imaginary amplitudes can be negative, and the constraints on those probabilities demand that any net probability be real and nonnegative, but the contributions can be negative-- and that is what never happened before quantum mechanics.

 

[PhilDSP]

1. To each dynamical variable there is a Hermitian operator whose egienvalues are the possible values of the dynamical variable.

2. To each state there corresponds a unique state operator P that must be Hermitian, non-negative and of unit trace. The average <A> for a dynamical variable with operator A in the virtual ensemble of events that may result from the preparation procedure for that state is <A> = Tr(PA).

Those are nice and economical. But even they seem to entangle the mathematical tools to some extent with the motivation and perspective of research and practice of QM. The tools may change with time or the situation, but what about the motivation of QM is worthy of definition?

 

[bhobba]

but what about the motivation of QM is worthy of definition

QM is extensively used in applications such as for example understanding how transistors work.

Thanks
Bill

 

[bhobba]

In my opinion complex numbers are an unnecessary complication, but what they are representing is essential. I like the approach Feynman took in QED, where as far as I recall he never mentions complex numbers but instead uses the simple geometrical idea of a clock with one hand. One may model amplitudes and their cancellations in this simple and intuitive geometric way.

Yes it does - that turning arrow he talks about is complex numbers in disguise. But you are correct - it is not explicitly stated and gives a very nice intuitive picture of what is going on independent of the math and explains why complex numbers are used.

Thanks
Bill