About the mathematical structure of wave functions

[burakumin]

Hello everybody

Two questions concerning the notion of wave function.

1. Every quantum mechanics book describes the space of wave functions (or more generally of quantum states) of a given system as a Hilbert space. But correct me if I'm wrong QM also says that two elements of such a Hilbert space describe the same state provided they are collinear (i.e. provided that one is the product of the other by any non zero complex number). I interpret this property as an indication that the difference between such a pair of wave functions is not physical but is merely a modelling artifact. In that condition why do quantum mechanicians keep representing wave functions as elements of vector spaces rather than of projective spaces. There is a small article on wikipedia on projective Hilbert spaces but without any reference. What is the real value of the vector space structure if it introduces non-physical (and then useless) information?

2. Let A and B be two Galilean frames of reference with respective points of origin $O_A$ and $O_B$ (supposed equal at t = 0) and constant relative vector speed $\overrightarrow{v_{BA}}$. Let us consider a particle the wave function of which is $\phi_A$ in frame A and $\phi_B$ in frame B. We have at a given time, for a point Q of space:
$\phi_B(Q) = \exp\left (\frac{2 \pi \imath}{h} m \left(\overrightarrow{O_BQ} - \frac{\overrightarrow{O_BO_A}}{2}\right) \cdot \overrightarrow{v_{BA}}\right ) \phi_A(Q)$
This means the usual notion of wavefunction is an entity dependent on the frame of reference.

Just as a tensor can be described either as a table of numbers with respect to a certain base or more abstractly as a multilinear object independent of bases, has anyone tried to use some more abstract mathematical object for wavefunction such that its projection/coordinates with respect to a certain Galilean frame yields the classical wave function ? The presence of the exponential makes me think about the possible use of some finite dimensional algebra but I couldn't find something that fits well.

Thank you

 

[dextercioby]

1. The space of pure states is a projective Hilbert space, i.e. the space of all possible (unit) rays of a complex inf-dim separable Hilbert space. To describe dynamics though, we resort to representatives, that is to a generic (unit norm) vector from each possible ray. Mixed states are properly described through von Neumann statistical operators.
2. Yes, the wavefunction is dependent of the frame of reference, simply because the relationship between different frames dictates which states are achieveable and their dynamics through the notion of symmetry: symmetry groups or more generally dynamical groups.

 

[burakumin]

Hi dextercioby,

> 1. The space of pure states is a projective Hilbert space, i.e. the space of all possible (unit) rays of a complex inf-dim separable Hilbert space. To describe dynamics though, we resort to representatives, that is to a generic (unit norm) vector from each possible ray.

But precisely why representatives would be any better than using directly rays. Vectors enable many operations that are non-sense in projective spaces. For example the null vector is a central notion of the vector space framework whereas it typically has no equivalent in projective spaces.

> 2. Yes, the wavefunction is dependent of the frame of reference, simply because the relationship between different frames dictates which states are achieveable and their dynamics through the notion of symmetry: symmetry groups or more generally dynamical groups.

Sorry but I don't see how this answers my question. I asked about the possibility of modeling wave functions with frame-independent entities.

For example I was wondering about how would it be possible to write the above equation as:
$\phi_B(Q) = \frac{f_A(Q)}{f_B(Q)} \phi_A(Q)$
which would allow me to consider $f_B(Q) \phi_B(Q) = f_A(Q) \phi_A(Q)$ as the absolute wave function. One can for example define formally:
$f_A(Q) = \exp\left (\frac{2 \pi \imath}{h} m \left( Q - \frac{1}{2} O_A \right) \cdot \overrightarrow{v_A}\right)$
which formally works ! But this requires to give a proper and consistant meaning to all those operations and objects. What is for example half a point ? (I cannot equate a point to a displacement vector without defeating my whole attempt to find frame-independent entities). Yet, there are solutions for this, like using vector embeddings of affine spaces. I can also consider $\overrightarrow{v_A}$ as an absolute speed in a spacetime framework. I can try to use Clifford-like algebras to multiply vectors, but the presence of exponential seems to force me to use finite dimensional algebras only. So in the end I couldn't find a consistant framework.

The attempt considered above might be a dead-end, but I'd like to know if some attempts of this kind were known.

 

[dextercioby]

The universal principle of quantum mechanics is the superposition one. You cannot superpose rays, you can superpose vectors, so that the vector space structure is mandatory.

 

[Jazzdude]

1. Every quantum mechanics book describes the space of wave functions (or more generally of quantum states) of a given system as a Hilbert space. But correct me if I'm wrong QM also says that two elements of such a Hilbert space describe the same state provided they are collinear (i.e. provided that one is the product of the other by any non zero complex number).

This is accurate so far. But the interesting question really is, why such vectors are physically equivalent. I'll come back to this.

I interpret this property as an indication that the difference between such a pair of wave functions is not physical but is merely a modelling artifact.

No, it's not really a modeling artifact. We have good reasons to use the linear space representation. Let's start with the fact that superpositions are very important both for understanding the theory and for working with it. Also, the time evolution of states is easy to formulate as a linear differential equation. But the third reason is really the most compelling (at least to me): The projective structure already completely follows from the linear structure without the need for additional postulates. Because the evolution of the state is entirely linear, there's no structural difference in the histories produced by collinear states. Or in other words, scalar (i.e. C\{0}) multiplication commutes with the time evolution and is a bijection. If you take the quotient of the linear space with this equivalence relation you get the projective space you refer to. So it is an emergent structure.

If you want to start with the projective space (an approach that is usually called geometric quantum theory) you get the structure of a Kähler manifold and a symplectic dynamic structure on top of it. This is a workable approach, but you have to postulate more because the structure doesn't emerge naturally anymore from linearity.

Note however that the quotient space or projective space is not the end of the reduction process. It's surely not correct to say that the rays are the true physical states, because there are more bijections on the representation space that commute with the dynamics. This becomes especially relevant when the information about the quantum state is restricted. If you are interested in the consequences of following this approach rigorously then please see my paper. It's not an easy read, but I believe you will find what you're asking for in there.

2. Let A and B be two Galilean frames of reference ...
Just as a tensor can be described either as a table of numbers with respect to a certain base or more abstractly as a multilinear object independent of bases, has anyone tried to use some more abstract mathematical object for wavefunction such that its projection/coordinates with respect to a certain Galilean frame yields the classical wave function ?

Yes, that more abstract representation you refer to is the baseless Dirac representation. The problem you describe only appears because you impose the artificial choice of a spatial basis onto the quantum state by expanding it in that basis. Or have I misunderstood your question?


Cheers,

Jazz

 

[burakumin]

Hello Jazz

No, it's not really a modeling artifact [...]

If you want to start with the projective space (an approach that is usually called geometric quantum theory) you get the structure of a Kähler manifold and a symplectic dynamic structure on top of it. This is a workable approach, but you have to postulate more because the structure doesn't emerge naturally anymore from linearity.

Note however that the quotient space or projective space is not the end of the reduction process. It's surely not correct to say that the rays are the true physical states, because there are more bijections on the representation space that commute with the dynamics. This becomes especially relevant when the information about the quantum state is restricted. If you are interested in the consequences of following this approach rigorously then please see my paper. It's not an easy read, but I believe you will find what you're asking for in there.

First thank you very much Jazz for all this detailed information. I have no real knowledge about Kälher Manifold but generally I like to discover this kind of abstract structures. I have also started to read your article. I guess that I first need to give a deeper look at all this before coming back with precise and relevant questions.

Yes, that more abstract representation you refer to is the baseless Dirac representation. The problem you describe only appears because you impose the artificial choice of a spatial basis onto the quantum state by expanding it in that basis. Or have I misunderstood your question?

I don't think we're exactly talking about the same thing. I'm not specially referring to some 3 dimensional basis of space (which I believe is not really difficult to get rid of) if this is what you had in mind. It is more about the possibility of removing from the usual wave functions:
- the explicit use of a (galilean) frame (that is to say of a given direction in spacetime that correspond to arbitrary and relative notion of rest) ;
- the explicit use of an origin to space.
However, to make a link with your comment, I have the intuition that these two arbitrary choices are equivalent to the choice of some arbitrary coordinates (or to some basis vectors) of some abstract structure. This is precisely this structure I'd like to surface.

 

[strangerep]

Burakumin,

Maybe I misunderstand where you're coming from, but I get the impression you think of the wave function as an entity that "lives" in spacetime. If so, that's the source of a lot of your difficulties.

In fact, the Hilbert space of (non-rel) QM is constructed by insisting that it carry a unitary representation of the Galilei group. I.e., the elements of the (abstract) Galilei group must map to operators on the Hilbert space. So, in a sense, the entire Hilbert space is the more general structure you seem to be looking for.

As a simpler exercise, perhaps consider just SO(3), the group of spatial rotations. The construction of (concrete) Hilbert spaces corresponding to the various unitary irreducible representations of SO(3) is standard material is most QM books -- see quantization of angular momentum. For example, Ballentine ch7 (iirc). In fact, if you haven't studied Ballentine's QM textbook, I'm sure it would be worth the investment of your time and money...

 

[burakumin]

Hello strangerep

Maybe I misunderstand where you're coming from, but I get the impression you think of the wave function as an entity that "lives" in spacetime. If so, that's the source of a lot of your difficulties.

Well, I see the classical wave function (of a single particle) as a complex valued function defined on a 3 dimensional "spatial" vector space associated to a Galilean frame and that depends on time. This leads me to believe one can abstract all this by embedding this object (or rather all these objects for each t of time) in a Galilean spacetime ( loosely speaking a 1 dimensional time space with a 3 dimensional fiber bundle) so that I don't need to think about it as related to a peculiar frame with a peculiar origin. Is that complete nonsense?

In fact, the Hilbert space of (non-rel) QM is constructed by insisting that it carry a unitary representation of the Galilei group. I.e., the elements of the (abstract) Galilei group must map to operators on the Hilbert space. So, in a sense, the entire Hilbert space is the more general structure you seem to be looking for.

Maybe my error is to keep focusing on the wave function instead of considering the abstract Hilbert space but I believed this space and the space of (positional) wave function were isomorphic. Am I wrong ? If it is correct, it means that if we can think about the abstract state vector as some "coordinate free" entity (that can undergo active transformations by action of the Galilean group for example) it should also be possible to describe the wave function as a "coordinate free" entity with the same properties, shouldn't it ?

As a simpler exercise, perhaps consider just SO(3), the group of spatial rotations. The construction of (concrete) Hilbert spaces corresponding to the various unitary irreducible representations of SO(3) is standard material is most QM books -- see quantization of angular momentum. For example, Ballentine ch7 (iirc). In fact, if you haven't studied Ballentine's QM textbook, I'm sure it would be worth the investment of your time and money...

That might be the best way but honestly, given what I've seen in a first glance (http://www-dft.ts.infn.it/~resta/fismat/ballentine.pdf), the approach seems to be the regular matrix stuff. Matrices have always been my enemies in physics. To me, coordinates deeply obfuscate structures by inserting useless arbitrary data and by making everything look similar (A disaster when I tried to understand spinors for example). As complex as it may be, I'm far more attracted by trying first to understand Jazz's article even if it eventually turns out to be too difficult for my level. But I will try to give another glance at your proposition.

 

[strangerep]

Well, I see the classical wave function (of a single particle) as a complex valued function defined on a 3 dimensional "spatial" vector space associated to a Galilean frame and that depends on time. This leads me to believe one can abstract all this by embedding this object (or rather all these objects for each t of time) in a Galilean spacetime ( loosely speaking a 1 dimensional time space with a 3 dimensional fiber bundle) so that I don't need to think about it as related to a peculiar frame with a peculiar origin. Is that complete nonsense?

What do you mean by "classical" wave function. Aren't we talking about QM here?

Maybe my error is to keep focusing on the wave function instead of considering the abstract Hilbert space but I believed this space and the space of (positional) wave function were isomorphic. Am I wrong ?

Thinking of the wave function as an object living in spacetime causes difficulties when one considers multi-particle situations. It turns out that thinking of 2 particles as having wave functions defined on the same underlying spacetime is untenable. One must instead construct a tensor product of Hilbert spaces. Ballentine covers this quite well.

If it is correct, it means that if we can think about the abstract state vector as some "coordinate free" entity (that can undergo active transformations by action of the Galilean group for example) it should also be possible to describe the wave function as a "coordinate free" entity with the same properties, shouldn't it ?

Think about what makes a scalar, spinor, vector, 2-tensor, etc, fundamentally different types of quantities, i.e., distinct types independent of any underlying coordinate system. What makes them different? A low-brow answer is that their components transform differently under the action of the group of symmetries applicable to the underlying space (here, Galilei or Poincare). A more advanced answer is that they correspond to different irreducible representations ("irreps") of that symmetry group. A significant difference between classical and quantum is then simply that we seek a space (a Hilbert space) which carries one of these irreps unitarily.

I'm guessing you don't know what the terms "representation", "carries" and "unitarily" mean, as I used them above? But this is critical to understanding quantum theory (and indeed elementary particle classification). I suggested Ballentine because his treatment of quantum angular momentum provides a reasonably gentle introduction to these concepts from a physicist-friendly perspective.

Advanced treatments of QFT (e.g., Weinberg) construct QFT essentially by analyzing the (unitary) irreps of the Poincare group. Acquiring a good understanding of these concepts is mandatory for any serious student of physics.

That might be the best way but honestly, given what I've seen in a first glance (http://www-dft.ts.infn.it/~resta/fismat/ballentine.pdf), the approach seems to be the regular matrix stuff.

My suggestion is to study Ballentine, not merely skim it. I don't think you'll find a really good answer to your original question until you understand mainstream QM more deeply.

Matrices have always been my enemies in physics.

If you don't make friends with matrices and linear algebra in general, your study of physics will remain severely handicapped.

 

[strangerep]

And... after a night's sleep it occurred to me that maybe I'm making this more difficult that it needs to be... :uhh:

If you don't like working with wavefunctions $\psi(x)$ with explicit coordinate dependence, then why not just use the Dirac bra-ket formalism, i.e., $|\psi\rangle$ ?

 

[burakumin]

Very unfortunately (and certainly my fault) I feel that the focus of this discussion is diverving from my initial question. So I'm going to try to separate questions about physics from the rest.

About some different approaches in physics (or in any topic that heavily relies on maths)

If you don't make friends with matrices and linear algebra in general, your study of physics will remain severely handicapped.

I had written a long answer about this remark, fearing that you were supposing I didn't know how matrices work. But apparently you do understand that my problem is with coordinate approach. Five years ago I might have entered a long debate with you about why coordinate-free approach is simpler. Because I do find it simpler:

• I think coordinates are by definition purely arbitrary because 90% of time there is no physical reason to prefer this set of coordinate rather than this one expect for maybe ad hoc pragmatical simplicity of computation.
• I think that using coordinates is transforming everything in numbers. Suddenly all begins to look similar which is super confusing. To me understanding is about making distinctions.

If we had done such a debate you may have vehemently disagree. And after 20 messages we would have just ended the conversation and forgotten my initial question.

Today I have at least learned that people can agree to disagree. I can accept that there are people (apparently the vast majority of physics teachers) that consider that coordinate approaches are far more intuitive and I'm perfectly fine if you consider Ballentine to be one of the best introductions to QM. But I also expect the same about these people: that they accept that other approaches might look simpler for some people. I don't want any "this is the only way" again when it turned out to be so wrong for other topics (for example general relativity or analytical mechanics). However I am not deeply opposed to matrices and coordinates as long as:

• they are introduced after some more abstract entities used for general explanations (and related to them);
• they are only presented as handy context-dependent calculation tools.

Unfortunately for me this is seldom the case.

If you don't like working with wavefunctions $\psi(x)$ with explicit coordinate dependence, then why not just use the Dirac bra-ket formalism, i.e., $|\psi\rangle$ ?

Now, this remark is perfectly valid. I would say:

• Because I'm curious about this question and about the possibility to define interesting geometrical objects with physical significance.
• Because I have never read clear explanations about how the abstract hilbert space $\mathcal{H}$ can be related with the notion of spacetime although some of your comments have been insightful.
• Because independence to coordinates of objects in $\mathcal{H}$ does not imply that wavefunctions must necessarily be coordinate-dependent.

Acquiring a good understanding of these concepts is mandatory for any serious student of physics.

And to conclude about this I am not a student of physics. I am a software engineer that tries to understand during his unfortunalety limited free time quantum mechanics, Schopenhauer's philosophy, second-order logic, cohomology theory, etc. Of course knowledge is certainly not free. Time and effort are required anyway. But if I can find approaches that fit me better it's always time saved for something else. Don't you think so?

About vocabulary

I'm guessing you don't know what the terms "representation", "carries" and "unitarily" mean, as I used them above? But this is critical to understanding quantum theory (and indeed elementary particle classification).

To my knowledge, a representation of a group G is any pair $(V,\phi)$ of a vector space and a group morphism $\phi : G \rightarrow GL(V)$. A representation is unitary if V is Hilbert and the image of $\phi$ belongs to the unitary group of V. I don't know what you specifically mean by "carries". That the representation is faithful? One of my last readings was Introduction to representation theory if that can reassure you. But to be honest and humble, it is correct that I don't have a fluent and perfect understanding about all representation theory as exposed in the latter document (and that I have skipped some chapters).

About my question about wave functions

A more advanced answer is that they correspond to different irreducible representations ("irreps") of that symmetry group. A significant difference between classical and quantum is then simply that we seek a space (a Hilbert space) which carries one of these irreps unitarily.

Ok so let me rephrase my question by trying to link it to representation theory (maybe stupidly but I least I've tried). To be clear, no einsteinian relativity here and just a single particle.

First let us define a galilean spacetime (not my invention). It is a 4d real affine space $\mathcal{A}$ with underlying vector space $\mathcal{V}$. $\mathcal{V}$ is equipped with a non zero linear form $\chi$ and $\mathrm{ker}\chi$ is equipped with an inner product that makes it a 3d Euclidean space. Let us call galilean group (or galilei group if you prefer) $\mathcal{G}$ the set of affine isomorphisms that preserve this structure. Besides, let us notice that $\chi$ induces a natural partition of $\mathcal{A}$ by 3d affine spaces called $\mathcal{A}_t, t\in \mathbb{R}$.

What do you mean by "classical" wave function. Aren't we talking about QM here?

By classical I meant clumsily "as defined in standard QM books". That is to say (correct me if this is wrong), for a given galilean frame, with specified axes and origin, a function of $\mathcal{L}_2( \mathbb{R}^3, \mathbb{C} )$ that depends on time but with constant norm.

My question was: can we try to mix those two concepts by defining on $\mathcal{A}$ a field of thingumabobs (thingumabob remaining to be defined) so that the projection of this field on a given frame-with-origin-and-axes yields the wave function above?

Supposing that thingumabobs are themselves a Hilbert space $\Theta$, then at given t, the space of fields $\mathcal{L}_2(\mathcal{A}_t, \Theta)$ with product $\langle \theta_1, \theta_2 \rangle = \int_{e \in \mathcal{A}_t} \langle \theta_1(e), \theta_2(e) \rangle de$ is also Hilbert. And then the space $\mathcal{B}$ of thingumabobs field on $\mathcal{A}$ with constant norm on every $\mathcal{A_t}$ is also.

Suppose one have a unitary representation $\phi$ of group $\mathcal{G}$ on $\Theta$. Then $\psi$ defined by:

$\psi(g) \cdot b : e \in \mathcal{A} \mapsto\phi(g) \cdot b( g^{-1} \cdot e) \in \Theta$​

with $b \in \mathcal{B}$ and $g \in \mathcal{G}$ seems to be a unitary representation on $\mathcal{B}$. However it is right I have not considered the question of irreducibility of such representations.

Once again I don't say thingumabobs must exist. This is precisely my question.

It turns out that thinking of 2 particles as having wave functions defined on the same underlying spacetime is untenable

Correct. But I have never said I didn't consider tensor product.

 

[strangerep]

[...] I can accept that there are people (apparently the vast majority of physics teachers) that consider that coordinate approaches are far more intuitive and I'm perfectly fine if you consider Ballentine to be one of the best introductions to QM. But I also expect the same about these people: that they accept that other approaches might look simpler for some people. I don't want any "this is the only way" again when it turned out to be so wrong for other topics (for example general relativity or analytical mechanics). However I am not deeply opposed to matrices and coordinates as long as:

• they are introduced after some more abstract entities used for general explanations (and related to them);
• they are only presented as handy context-dependent calculation tools.

Unfortunately for me this is seldom the case.

I have no problem with any of this, and I think both approaches are useful in different contexts.
Indeed, I made an effort to become "bilingual" some time ago.

[*] Because I have never read clear explanations about how the abstract hilbert space $\mathcal{H}$ can be related with the notion of spacetime although some of your comments have been insightful.

Ballentine answers this question quite extensively.

[*] Because independence to coordinates of objects in $\mathcal{H}$ does not imply that wavefunctions must necessarily be coordinate-dependent.

Is there a typo in that sentence? It doesn't make sense to me.

[...] I am not a student of physics. I am a software engineer that tries to understand during his unfortunalety limited free time quantum mechanics, [...]

Well, just because you're not studying the subject in a formal course doesn't mean you're not a "student".

Of course knowledge is certainly not free. Time and effort are required anyway. But if I can find approaches that fit me better it's always time saved for something else. Don't you think so?

For many years, I was exclusively a "coordinates and indices" person, and stayed away from coordinate-free approaches. Eventually, I came to understand that both approaches have their strengths and weaknesses, so I made an effort to become more bilingual.

By classical I meant clumsily "as defined in standard QM books". That is to say (correct me if this is wrong), for a given galilean frame, with specified axes and origin, a function of $\mathcal{L}_2( \mathbb{R}^3, \mathbb{C} )$ that depends on time but with constant norm.

My question was: can we try to mix those two concepts by defining on $\mathcal{A}$ a field of thingumabobs (thingumabob remaining to be defined) so that the projection of this field on a given frame-with-origin-and-axes yields the wave function above?

This sounds just like the usual $\psi(x) = \langle x|\psi\rangle$. See, e.g., ch4 of Ballentine.

That's why I suggested the Dirac formalism may be more to your liking. I don't see how it's really any different (fundamentally) from what you want.

All I can say is: have a go at reading+studying Ballentine carefully and slowly, from the beginning. Try to get past your initial distaste for his physicist-friendly notation and become more bilingual. (But this is the last time I'll repeat this advice.)