"It is the outer product of direct space and reciprocal space." You lied to me! :D Is the Hilbert space of the wave function then not a phase space? I think I recall that Feynman says something about the wavefunction not really working like a vector but a Hilbert space _is_ a vector space.
@Narf Whals yes, the Hilbert space of the wave function is absolutely _not_ the same thing as phase space. Remember that in QM, we can't know both the position and momentum of a particle, so defining a point in phase space makes absolutely no sense to do in quantum mechanics, phase space is purely classical. I don't exactly know the context where Feynman says that the wavefunction doesn't work like a vector, but I would guess that he means that the wavefunction can be a continuous function instead of just a discrete vector and that is why it is different, but you are exactly right, the Hilbert space is certainly a vector space, it just happens that it can be spanned by functions instead of vectors.
@@zapphysics Maybe youtube just really doesn't like me linking the Feynman lectures. This is my third time writing this... I think this is what I was remembering from Volume 3 chapter 8. "There is one minor difference between Eq. (8.1) and the dot product. We have that ⟨ϕ|χ⟩=⟨χ|ϕ⟩∗. But in vector algebra A⋅B=B⋅A. With the complex numbers of quantum mechanics we have to keep straight the order of the terms, whereas in the dot product, the order doesn’t matter. " But I think that just means he was comparing to a real vector space. In a complex one this just works. And I think my confusion about Phase Space and Hilbert Space comes from me thinking Configuration Space and Phase Space were the same thing. Sean Carroll often refers to Configuration Space when talking about the evolution of fields. Is _that_ Hilbert Space? Physics is very spacious.
@@narfwhals7843 Ah yes, you were probably getting trapped in youtube's spam filter with outside links. But yes, I agree with what you are saying about Feynman's lectures that he is probably just saying complex vectors behave differently from real vectors. As for configuration space, this one is a bit more tricky. As I understand, configuration space is essentially "half" of phase space: it is only the space made up of coordinates, not coordinates + momenta (keep in mind that "coordinate" is very general and also can include field variables as well). This tends to be helpful in quantum mechanics because path integrals usually run over all of configuration space, even those regions which do not satisfy the equations of motion. Whether or not this is a Hilbert space probably depends on the specific system you care about. For example, if I am looking at the classical solutions to a non-linear equation of motion, then the vector algebra is not guaranteed to be satisfied: if v1 and v2 are solutions to the eom, then v1 + v2 is not necessarily a solution due to the non-linearity. This is something we run into with GR: the field equations are highly non-linear, so we can't just find e.g. a two black hole solution by super-imposing two Schwarzschild solutions on top of each other. However, in linear cases, I would guess that solutions do often form Hilbert spaces (or at least, vector spaces) though one would need to study functional analysis a bit more than I have (that is to say, at all) to be able to correctly classify them.
@@zapphysics Thanks you! The linearity argument is interesting and reminds me of something I've been thinking about lately. Do we have any "right" or reason to expect nature to be linear at a fundamental level? Much of QM relies on this and as I understand that is a big part of the problem of unifying GR and QFT. Should we expect the Schrödinger equation to be a low energy approximation(are the string theory equations linear?) or do we expect the non linearity to arise from the linearity in some limit? We can linearize gravity for weak fields, maybe that's what QFT is? Or would that just be hiding our problems in a regime we can't test?
@@narfwhals7843 So this question has puzzled me a bit and I have been thinking about it to try to come up with a good answer. Unfortunately, I still don't think I have one, but I will try my best. I think that, in the most general cases a quantum theory will yield a non-linear theory in the classical limit. This is simply because we know that the quantum theory yields the classical field equations in this limit and for pretty much any interacting field theory, these field equations will be highly non-linear. As far as I understand, this goes for string theory as well. The string states (I am pretty sure) are linear and can be superposed and coherent string states form classical fields, one of which, in the low energy limit, behaves like the gravitational field from GR. So do we expect this to be "fundamental" i.e. should nature always prefer linearity over non-linearity? As far as I know, there isn't really a universally accepted answer. In fact, there is even a lot of research which goes into asking whether or not quantum mechanics is truly linear (there is a non-linear Schrödinger equation, for example), so I don't think that you will find any sort of definite answer. I personally am of the philosophy that it would be nice if nature, at its core, was as simple as possible: linear, highly symmetric (supersymmetric, conformally invariant, etc.), with only a couple of necessary fundamental objects (whether they be fields or something more abstract), and only through a long chain of symmetry breaking a low-energy effective theories do we end up at the messiness we see today. But like I said, this is a personal philosophy, not a rigid physical fact, and many people will disagree with this.
For the channel this good, the number of subscribers is very few
It starts at 4:44
"It is the outer product of direct space and reciprocal space." You lied to me! :D
Is the Hilbert space of the wave function then not a phase space? I think I recall that Feynman says something about the wavefunction not really working like a vector but a Hilbert space _is_ a vector space.
@Narf Whals yes, the Hilbert space of the wave function is absolutely _not_ the same thing as phase space. Remember that in QM, we can't know both the position and momentum of a particle, so defining a point in phase space makes absolutely no sense to do in quantum mechanics, phase space is purely classical. I don't exactly know the context where Feynman says that the wavefunction doesn't work like a vector, but I would guess that he means that the wavefunction can be a continuous function instead of just a discrete vector and that is why it is different, but you are exactly right, the Hilbert space is certainly a vector space, it just happens that it can be spanned by functions instead of vectors.
@@zapphysics Maybe youtube just really doesn't like me linking the Feynman lectures. This is my third time writing this...
I think this is what I was remembering from Volume 3 chapter 8.
"There is one minor difference between Eq. (8.1) and the dot product. We have that
⟨ϕ|χ⟩=⟨χ|ϕ⟩∗.
But in vector algebra
A⋅B=B⋅A.
With the complex numbers of quantum mechanics we have to keep straight the order of the terms, whereas in the dot product, the order doesn’t matter. "
But I think that just means he was comparing to a real vector space. In a complex one this just works.
And I think my confusion about Phase Space and Hilbert Space comes from me thinking Configuration Space and Phase Space were the same thing. Sean Carroll often refers to Configuration Space when talking about the evolution of fields. Is _that_ Hilbert Space?
Physics is very spacious.
@@narfwhals7843 Ah yes, you were probably getting trapped in youtube's spam filter with outside links. But yes, I agree with what you are saying about Feynman's lectures that he is probably just saying complex vectors behave differently from real vectors.
As for configuration space, this one is a bit more tricky. As I understand, configuration space is essentially "half" of phase space: it is only the space made up of coordinates, not coordinates + momenta (keep in mind that "coordinate" is very general and also can include field variables as well). This tends to be helpful in quantum mechanics because path integrals usually run over all of configuration space, even those regions which do not satisfy the equations of motion. Whether or not this is a Hilbert space probably depends on the specific system you care about. For example, if I am looking at the classical solutions to a non-linear equation of motion, then the vector algebra is not guaranteed to be satisfied: if v1 and v2 are solutions to the eom, then v1 + v2 is not necessarily a solution due to the non-linearity. This is something we run into with GR: the field equations are highly non-linear, so we can't just find e.g. a two black hole solution by super-imposing two Schwarzschild solutions on top of each other. However, in linear cases, I would guess that solutions do often form Hilbert spaces (or at least, vector spaces) though one would need to study functional analysis a bit more than I have (that is to say, at all) to be able to correctly classify them.
@@zapphysics Thanks you!
The linearity argument is interesting and reminds me of something I've been thinking about lately. Do we have any "right" or reason to expect nature to be linear at a fundamental level? Much of QM relies on this and as I understand that is a big part of the problem of unifying GR and QFT. Should we expect the Schrödinger equation to be a low energy approximation(are the string theory equations linear?) or do we expect the non linearity to arise from the linearity in some limit?
We can linearize gravity for weak fields, maybe that's what QFT is? Or would that just be hiding our problems in a regime we can't test?
@@narfwhals7843 So this question has puzzled me a bit and I have been thinking about it to try to come up with a good answer. Unfortunately, I still don't think I have one, but I will try my best. I think that, in the most general cases a quantum theory will yield a non-linear theory in the classical limit. This is simply because we know that the quantum theory yields the classical field equations in this limit and for pretty much any interacting field theory, these field equations will be highly non-linear. As far as I understand, this goes for string theory as well. The string states (I am pretty sure) are linear and can be superposed and coherent string states form classical fields, one of which, in the low energy limit, behaves like the gravitational field from GR.
So do we expect this to be "fundamental" i.e. should nature always prefer linearity over non-linearity? As far as I know, there isn't really a universally accepted answer. In fact, there is even a lot of research which goes into asking whether or not quantum mechanics is truly linear (there is a non-linear Schrödinger equation, for example), so I don't think that you will find any sort of definite answer. I personally am of the philosophy that it would be nice if nature, at its core, was as simple as possible: linear, highly symmetric (supersymmetric, conformally invariant, etc.), with only a couple of necessary fundamental objects (whether they be fields or something more abstract), and only through a long chain of symmetry breaking a low-energy effective theories do we end up at the messiness we see today. But like I said, this is a personal philosophy, not a rigid physical fact, and many people will disagree with this.
Could have been broken into less intimidating chunks maybe