I Vector Operators in Quantum Mechanics: Adapting to Different Coordinate Systems

[David Olivier]

I have always seen “vector” operators, such as the position operator $\vec R$, defined as a triplet of three “coordinate” operators; e.g. $\vec R = (X, Y, Z)$. Each of the latter being a bona fide operator, i.e. a self-adjoint linear mapping on the Hilbert space of states $\mathcal H$.

(I am putting aside here completely all the subtleties discussed in a recent thread.)

I'd like to be able to express the object $\vec R$ without a reference to a particular coordinate system.

What could such an operator be? A linear mapping $\mathcal H \to \mathcal E \otimes \mathcal H$, with $\mathcal E$ the vector space of the three-dimensional “physical” space? This would allow us to write $\vec R | \vec r > = \vec r \otimes | \vec r >$

However, $\mathcal E$ is normally a real vector space, while $\mathcal H$ is a complex one. Perhaps it will do if we first convert $\mathcal E$ into a complex vector space?

Operators are to be self-adjoint. How do we express this condition for such a vector operator?

What about “pseudovectors” such as the angular momentum operator? Would we view them as antisymmetric elements of $\mathcal E \otimes \mathcal E$?

 

[andrewkirk]

It looks like in your post $\mathcal E$ denotes $\mathbb R^3$. If so then, unless I have misunderstood your question, $\mathcal E$ plays no role in the definition. A three-dimensional position operator is a linear map from $\mathcal H$ to $\mathcal H$ where $\mathcal H=\mathcal H_0\otimes\mathcal H_0\otimes\mathcal H_0$ and $\mathcal H_0$ is an infinite-dimensional Hilbert space that is isomorphic to a suitably-restricted subspace of $W$, the space of complex distributions on $\mathbb R$.

$W$ is a bit like the space of all functions from $\mathbb R$ to $\mathbb C$ but, in addition to an isomorphic copy of that space, it contains additional members that are not functions, such as the Dirac Delta.

The analog in $\mathcal H$ to the point $(x,y,z)\in \mathbb R^3$ is the tensor product
$$\delta_x\otimes\delta_y\otimes\delta_z$$
where $\delta_a$ is the Dirac Delta distribution that is 'zero every where except at $a$'.

 

[David Olivier]

Well, I'm trying to have a formulation that doesn't refer to a coordinate system. So $\mathcal E$ is a three-dimensional vector space, but I precisely don't want to call it $\mathbb R^3$, which would assume that I have determined the x, y and z axes.

If I understand you, your $\mathcal H_0$ is the one-dimensional spinless state space; which is consistent with a three-dimensional state space $\mathcal H = \mathcal H_0 \otimes \mathcal H_0 \otimes \mathcal H_0$. However, the position operator cannot be a linear map from $\mathcal H \to \mathcal H$. In the usual, coordinate system dependent, description, we have three position operators, $X$, $Y$ and $Z$, each of which is such a linear map; the combination of the three doesn't make a linear map $\mathcal H \to \mathcal H$, but might be seen as a linear map $\mathcal H \to \mathcal E \otimes \mathcal H$. This is what I was getting at in my OP.

 

[andrewkirk]

the combination of the three doesn't make a linear map

Why do you think it isn't a linear map?

Say $B$ is a basis for $\mathcal H_0$. Then the Cartesian product $B'=B\times B\times B$ is a basis for $\mathcal H$. The action of $X\otimes Y\otimes Z$ on an element $|x\rangle\otimes|y\rangle\otimes|z\rangle$ of the ($B$-induced) basis of $\mathcal H$, for $|x\rangle,|y\rangle,|z\rangle\in\mathcal B$, is

$$(X\otimes Y\otimes Z)(|x\rangle\otimes|y\rangle\otimes|z\rangle)
\triangleq X|x\rangle\otimes Y|y\rangle\otimes Z|z\rangle $$

and this is then extended to linear combinations of basis elements to make a linear map from $\mathcal H$ to itself.

IIRC it is reasonably straightforward to show that this map is consistent with each of the component maps.

 

[David Olivier]

Thanks for the answer, but I don't see that it works.

When I say that the position operator is not a linear map $\mathcal H \to \mathcal H$, the issue is not that it is not linear, but that the target set cannot be $\mathcal H$.

Let's consider the action of the solution you give on the ket $| \vec r_0 \rangle$representing a particle localized in $\vec r_0$ with coordinates $(x_0, y_0, z_0)$; in terms of wavefunction, this ket represents $\delta(\vec r - \vec r_0)$. Following your construction, we have $| \vec r_0 \rangle = | x_0 \rangle \otimes | y_0 \rangle \otimes | z_0 \rangle$, or, in terms of wavefunctions, $\delta(\vec r - \vec r_0) = \delta(x - x_0) \delta(y - y_0) \delta(z - z_0)$. But then the action of the operator you describe on this ket gives:

$(X \otimes Y \otimes Z) | \vec r_0 \rangle = x_0 y_0 z_0 | \vec r_0 \rangle$

Thus $X \otimes Y \otimes Z$ would be an operator with eigenvalue $x_0 y_0 z_0$ on $| \vec r_0 \rangle$ ‑ which is nothing like what we want, and doesn't even transform properly if we change our coordinate system.

What we need is for our eigenvalue to be a three-dimensonal vector, representing, for a localized particle, its position; which means that our three-dimensional position operator cannot be $X \otimes Y \otimes Z$ mapping $\mathcal H$ to itself, but must map $\mathcal H$ to some larger space, which I suggested might be $\mathcal E \otimes \mathcal H$, with $\mathcal E$ the three-dimensional ordinary vector space, perhaps made into a complex vector space.

 

[andrewkirk]

It looks like I've misunderstood what you are after. I'll have another try.

It sounds like you are trying to define a linear map between Hilbert spaces that corresponds to an instantaneous measurement of the 3D position of a particle, in the same way that a measurement operator (a linear map from $\mathcal H$ to $\mathcal H$) corresponds to the measurement of a scalar amount. Is that right?

If so, the problem that looms is that multiple measurements - at least in the QM I've seen - are never simultaneous. One measurement gives one result - a single real number. If we want to measure 3D position we need to make three separate measurements, and they will not be simultaneous, so they cannot be encapsulated in a single operator.

Consider, for example, the measurement of spin. A central theme in the Bell theorem is that we can only measure spin in relation to a given axis. We cannot make a single measurement that tells us a particle's pre-measurement 3D spin vector.

If it were possible to make three scalar measurements simultaneously - or to measure a 3D vector simultaneously, which is the same thing - I suspect the formulation of Bell's Theorem would be very diffferent from what it is.

 

[David Olivier]

It looks like I've misunderstood what you are after. I'll have another try.

Thanks.

It sounds like you are trying to define a linear map between Hilbert spaces that corresponds to an instantaneous measurement of the 3D position of a particle, in the same way that a measurement operator (a linear map from $\mathcal H$ to $\mathcal H$) corresponds to the measurement of a scalar amount. Is that right?

Quite right.

However, your objection ‑ that it is impossible to measure simultaneously the values of $x$, $y$ and $z$ ‑ does not hold. QM tells us that we cannot measure simultaneously the values of operators that do not commute. Because, for instance, $[X, P_x] = i$, one cannot measure both $x$ and $p_x$ together; there is no state that is an eigenstate for both operators $X$ and $P_x$. But the position operators do commute; for instance, $[X, Y] = 0$. One can measure simultaneously all three spatial coordinates, in other words the exact position of a particle (if indeed it is in such an exact position, which in principle is possible; that is what the eigenstate $| \vec r \rangle$ is about).

Your objection does apply, for instance, if I was attempting to form an operator for the angular momentum vector (or pseudovector), since, as you say for spin, the three coordinate operators $J_x$, $J_y$ and $J_z$ do not commute. It would not however keep us from wanting to form the corresponding vector “operator”, which would be a linear mapping, and have some property replacing the auto-adjoint requirement, but would have no eigenstates. This is indeed a difficulty, that I hadn't thought of. But in the case of position, it simply isn't there.

 

[andrewkirk]

The obstacle that I was imagining was practical rather than theoretical. I expect the two are related. It's not clear to me how, but non-commutativity was not what I had in mind.

The only two methods that immediately come to my mind to measure the 3D location of an object are triangulation and radar/sonar. Triangulation consists of two separate 1D measurements and radar/sonar is a rapid sequence of measurements as the beam sweeps out a sector. So in each case only 1D measurements are ever made.

Do you have an experimental procedure in mind whereby one would measure the 3D location of an object in an instant, where that measurement cannot be interpreted as a 1D measurement?

 

[David Olivier]

Photographic plates record photons one by one. When they record a photon, you have measured the three coordinates simultaneously. I don't see how that could be interpreted as a 1D measurement.

But this is irrelevant to my problem, which was: we have a certain notion, called “position”, that, in a given coordinate system, is expressed as three operators, $X$, $Y$ and $Z$. (Or alternatively, a notion called “angular momentum” that is expressed as three operators $J_x$, $J_y$ and $J_z$.) Why is it that one cannot express this notion independently from the coordinate system, using only the basic tools of the 3D vector space and other objects such as the Hilbert space that derive from it?

(By the way, to say that a vector space is 3D does not mean that we have chosen some coordinate system. It has the property of being 3D independently from any coordinate system. For instance, it is a vector space in which one can embed a Möbius strip but not a Klein bottle.)

 

[PeterDonis]

it is a vector space in which one can embed a Möbius strip but not a Klein bottle

That isn't a vector space, it's a metric space. Vector spaces have an inner product, but not a metric.

 

[David Olivier]

Vector spaces with an inner product are metric spaces.

 

[PeterDonis]

Vector spaces with an inner product are metric spaces.

They can be given a metric, induced by the norm that is derived from the inner product. But that metric is not logically required by the vector space or inner product.

A better way of defining the dimension of a vector space, which does not require using any metric space properties, is to count the number of vectors required to form a basis. This is also coordinate-independent.

 

[David Olivier]

Specifying the inner product is tantamount to specifying the metric.

Anyway, the issue of Möbius strips and Klein bottles isn't even a metric one. It is a topological issue, and on a finite-dimensional vector space there is but one topology that is compatible with the vector space structure. It is thus a property of the vector space itself whether you can or cannot embed in it a given structure such as a Möbius strip or a Klein bottle.

A better way of defining the dimension of a vector space, which does not require using any metric space properties, is to count the number of vectors required to form a basis. This is also coordinate-independent.

Specifying whether or not you can insert a Klein bottle or a Möbius strip does not require any metric properties.

Counting the number of vectors to form a basis is the obvious method. I only mentioned the question of what you can or cannot embed in the space to stress that the dimension of the space is part of the very fabric of the space independently from any choice of coordinates.

Anyway, this is a secondary issue. My initial question still stands: how can we define a “vector operator” without referring to a given coordinate system?

 

[vanhees71]

If the inner product is positive definite, it implies a metric, but there are metric vector space where the metric is not induced by an inner product.

 

[yossell]

It's unclear to me whether your worry is one about coordinate systems or vectors.

If you want to avoid using coordinate systems in expressing the laws, why does moving to a vector space help? What is `the' vector space of 3 dimensional space (that you mention in your first post)?

After all, which position a given vector picks out is still relative to a choice of origin. We would still need to make sure that the laws were invariant under the relevant transformations.

 

[andrewkirk]

Photographic plates record photons one by one. When they record a photon, you have measured the three coordinates simultaneously. I don't see how that could be interpreted as a 1D measurement.

What a spot on the plate tells us is that, at some time during the exposure period, a photon struck that spot. I don't see that as a 3D measurement, since it is implicit when we talk about measurements that we know the time at which the measurement is taken. With the plate we do not know the time.

 

[stevendaryl]

Getting back to the original question, it seems to me that in most treatments of quantum mechanics, only scalar-valued operators are considered. So we talk about L_z and L^2 but not \vec{L}.

Just thinking out loud (or at the keyboard)...

If you have a vector-valued operator \vec{V} and you apply it to a state |\psi\rangle, you don't get an element of the Hilbert space back, you get something like a triple of states: V_x |\psi\rangle, V_y |\psi\rangle, V_z |\psi\rangle. So is there a way to work with vectors that doesn't involve breaking it up into components?

Well, if you sandwich \vec{V} between two states, you get \langle \phi|\vec{V}|\psi \rangle. If \mathcal{E} is your vector space, then this will be an element of the complexification of \mathcal{E}. I don't know of there is a standard term for that, but let me call it \bar{\mathcal{E}}. So \vec{V} has type: \mathcal{H} \times \mathcal{H} \rightarrow \bar{\mathcal{E}}, and is anti-linear in the first argument and linear in the second argument. In tensor terms, I think that's isomorphic to \mathcal{H}^* \otimes \mathcal{H} \otimes \bar{E}. In terms of Dirac notation, we could write:

\vec{V} = \sum_{\alpha \beta j} C_{\alpha \beta j} \mathbf{e_j} |\psi_\alpha\rangle \langle \psi_\beta |

where C_{\alpha \beta j} = \langle \psi_\alpha|V_j|\psi_\beta \rangle, and where e_j is a basis vector for \mathcal{E}, and |\psi_\alpha\rangle is a complete orthonormal basis for the Hilbert space \mathcal{H}.

 

[vanhees71]

No, $\vec{L}$ is a vector operator, $L_z$ is a vector component and not a scalar. By definition a vector operator obeys the commutation relations
$$[L_i,V_j]=\mathrm{i} \epsilon_{ijk} V_k,$$
where $L_i$ are the components of $\vec{L}$ and $V_k$ the components of $\vec{V}$ with respect to an arbitrary Cartesian basis.

This implies that the expectation values of vector operators transform as vectors, as it should be.

All this of course generalizes to tensor operators of higher rank.

 

[David Olivier]

It's unclear to me whether your worry is one about coordinate systems or vectors.

If you want to avoid using coordinate systems in expressing the laws, why does moving to a vector space help? What is `the' vector space of 3 dimensional space (that you mention in your first post)?

After all, which position a given vector picks out is still relative to a choice of origin. We would still need to make sure that the laws were invariant under the relevant transformations.

Your point about the origin is a good one.

I am looking for an expression of the physical laws of quantum mechanics that doesn't depend on the choice of the three ordinary spatial directions. My vector space $\mathcal E$ is the ordinary three dimensional real (i.e. over field $\mathbb R$) vector space. I want only expressions with vectors from that vector space, time, and objects built on it (such as $\mathcal H = L^2[\mathcal E]$, the Hilbert space of states).

In classical mechanics, for instance, you have expressions such as $\vec f = m \vec a$, which involve vectors, not scalars. Of course, one can project this equation onto any coordinate system (i.e. choose a basis $(\vec i, \vec j, \vec k)$, and obtain three equations about scalars, but that is artificial and useless.

I would like an expression in quantum mechanics that, like those of classical mechanics, doesn't care about the basis you have chosen.

Checking that laws are invariant under various transformations is all very well, but it is better still if the relevant invariances were incorporated in the formalism itself. If you have expressions involving $x$, $y$, $z$, $f_x$ and so on, you may want to check that any physical results vary properly under rotation; but if you have expressions only with vectors, there is nothing to check.

Your point about the origin is a good one, because one would also want to express everything independently from a specific origin. For instance, expressing the position (in classical mechanics) by a position vector $\vec r$ is better than with a triplet $(x, y, z)$, but still implies an artificial choice of origin. It would be better still to have expressions directly in an affine space (the space of points). My question is about how to do the first step in quantum mechanics; leaving the second step for after.

Another issue is that of movement. It would be good to have expressions that don't depend on the choice between different Galilean frames of reference. But that too is an additional step, that can be left for later. For the moment, I would be happy if we could get rid of the pesky $(X, Y, Z)$.

 

[David Olivier]

What a spot on the plate tells us is that, at some time during the exposure period, a photon struck that spot. I don't see that as a 3D measurement, since it is implicit when we talk about measurements that we know the time at which the measurement is taken. With the plate we do not know the time.

If we have other kinds of detectors, we may know the time too, and have all three coordinates at a precise time. But as I said, this is not very relevant to my issue.

 

[vanhees71]

Sure, both Newtonian and specia-relativistic space time is isotropic and thus a fundamental Hamiltonian must be a scalar (operator) under rotations. That's the same in classical and quantum theory. Symmetry principles are valid for all of physics!

 

[David Olivier]

Well, if you sandwich ⃗VV→\vec{V} between two states, you get ⟨ϕ|⃗V|ψ⟩⟨ϕ|V→|ψ⟩\langle \phi|\vec{V}|\psi \rangle. If EE\mathcal{E} is your vector space, then this will be an element of the complexification of EE\mathcal{E}. I don't know of there is a standard term for that, but let me call it ¯EE¯\bar{\mathcal{E}}. So ⃗VV→\vec{V} has type: H×H→¯EH×H→E¯\mathcal{H} \times \mathcal{H} \rightarrow \bar{\mathcal{E}}, and is anti-linear in the first argument and linear in the second argument. In tensor terms, I think that's isomorphic to H∗⊗H⊗¯EH∗⊗H⊗E¯\mathcal{H}^* \otimes \mathcal{H} \otimes \bar{E}.

I was suggesting that our $\vec V$ could be something like (in your terms) $\bar {\mathcal E} \otimes \mathcal H$. Then $\langle \phi | \vec V | \psi \rangle$ would be, not an object of $\mathcal H \otimes \bar {\mathcal E} \otimes \mathcal H$, but of the contraction of this following the first and last vectors (the contraction amounts to performing the scalar product), that is an object of $\bar {\mathcal E}$, which is what we want.

 

[stevendaryl]

I was suggesting that our $\vec V$ could be something like (in your terms) $\bar {\mathcal E} \otimes \mathcal H$. Then $\langle \phi | \vec V | \psi \rangle$ would be, not an object of $\mathcal H \otimes \bar {\mathcal E} \otimes \mathcal H$, but of the contraction of this following the first and last vectors (the contraction amounts to performing the scalar product), that is an object of $\bar {\mathcal E}$, which is what we want.

I don't think that's right. I would say that \vec{V} |\psi\rangle would be of type \bar{\mathcal{E}} \otimes \mathcal{H}, but not \vec{V} by itself.

 

[David Olivier]

Sorry, I got it wrong. I meant (assuming I was in my right mind) that $\vec V$ should be a linear mapping from $\mathcal H$ to $\bar {\mathcal E} \otimes \mathcal H$, which is essentially what you are saying (and that I was saying in my OP, except that I initially omitted the fact that it should be $\bar {\mathcal E}$, not just $\mathcal E$).

(And I meant $\mathcal H^* \otimes \bar {\mathcal E} \otimes \mathcal H$, not $\mathcal H \otimes \bar {\mathcal E} \otimes \mathcal H$.)

 

[andrewkirk]

@David Olivier. Let's leave aside for now the question of whether an operator can correspond to an exact physical measurement, and consider it purely mathematically.

Operators are linear maps from the Hilbert space to itself. The Born rule tells us that each physical measurement corresponds to an operator and that:
1. After the measurement is performed, the system will be in the state whose ket is the projection of the pre-measurement ket onto one of the eigenspaces of $M$.
2. The value that is given by the measurement will be the eigenvalue of $M$ that corresponds to the eigenspace containing the post-measurement ket.

As I understand it, you want to find something that is similar to this, but for measuring a 3D vector rather than a scalar.

I think my original suggestion is the closest one can get to this. Another way to present that suggestion is to describe the operator, call it $X$, as $X_1X_2X_3$ where $X_i$ is the measurement of location coordinate $i$, given an orthogonal 3D spatial coordinate system with coordinates $x_1,x_2,x_3$. The $X_i$ all commute, so the order in which they are written does not matter.

We could think of this operator as corresponding to making the measurements $X_1,X_2,X_3$ in succession, so close after one another that system evolution between measurements can be ignored.

Considering the three measurements as one, we see that the Born rule, as presented above, is satisfied.

The result of the measurement is a scalar, equal to the product of the $x_i$s. However, given the measurements were actually performed individually in close succession, in practice we will have the three numbers. Further, we can describe them mathematically as:

• $x_i$ is the eigenvalue of $X_i$ corresponding to the eigenspace of $X_i$ that contains the post-measurement ket (here 'post-measurement' means post-all three measurements).

I think that's as close to the Born rule as we can get, given that operator eigenvalues are scalars.

Returning to the question of whether there is an actual physical measurement corresponding to the operator $X$, I am still convinced there is not. The closest we can come is to measure the three components in very rapid succession, relying on the mutual commutativity of the three component measurement operators for our hope that the measurements will not disturb one another.

 

[David Olivier]

Thanks for your answer, but I don't think it is close to what I am after. Your $X_1 X_2 X_3$ operator is inherently dependent on the choice of a basis. Its result ($x_1 x_2 x_3$) is not at all invariant under rotation, and actually doesn't give us the position of the particle (since knowing $x_1 x_2 x_3$ doesn't tell us the values of $x_1$, $x_2$ and $x_3$ separately).

I don't know why you think it so impossible to measure the 3D position of a particle. But why would that be relevant?

My suggestion (see discussion with stevendaryl above) is to consider the position operator $\vec R$ as a linear mapping from $\mathcal H$ to $\bar {\mathcal E} \otimes \mathcal H$. Clearly that needs to be fleshed out. But I don't see any objections, apart from the fact that it indeed is not a mapping from $\mathcal H$ to itself, so it is outside the rule that you state above, but that just means it is another kind of operator.

 

[vanhees71]

What a mess! It's really not that difficult. As any vector operator the three operators representing the Cartesian components of the position operator transform as a vector (that's why it's called a vector operator in the first place), i.e., for $R \in \mathrm{SO}(3)$ you have a unitary representation $\hat{U}(R):\mathcal{H} \rightarrow \mathcal{H}$ (such a unitary representation must exist since a Galileian invariant QT must admit rotations as symmetry operators). The operators for the Carstesian components of the position operator thus transform under rotations as
$$\hat{U}(R) \hat{\vec{x}} \hat{U}^{\dagger}(R)=R \hat{\vec{x}}.$$

 

[David Olivier]

What a mess! It's really not that difficult.

Sorry, vanhees71, but it's a mess mainly because almost no one has seriously attempted to answer the original question.

Your answer essentially adds nothing to what I had stated in the original post. I am not asking how to express the three Cartesian components of a vector operator in a particular basis of the 3D space, nor how they transform under $SO(3)$; I am asking how to express them without any choice of a basis in the first place.

It should really not be that difficult to read the question before answering it.

 

[andrewkirk]

I don't know why you think it so impossible to measure the 3D position of a particle. But why would that be relevant?

It's relevant because if , as I believe, we need to make three separate measurements, we make them from three specific, preferably orthogonal, directions. That makes an interaction between the measuring apparatus and the object, which has an impact on the post-measurement state. Because there are specific directions involved in the interaction, the post-measurement state will be a function of the directions from which the object was measured.

A perfect measurement in any direction $x$ will convert the wavefunction of the coordinate in that direction to a Dirac Delta. In practice we cannot make exact measurements, so the wavefunction will instead become a Gaussian, whose narrowness corresponds to the accuracy of the measurement.

If we perform three measurements from orthogonal directions in rapid succession, the post-measurement state in the position basis will be a distribution of a vector in $\mathbb R^3$ that has independent components with 1D Gaussian marginal. The rotational symmetries of this distribution will not be those of a sphere but of a cube that is aligned with the directions from which the measurements were performed.

If we rotate the axes from which we perform the measurements, in a way that is not a rotational symmetry of a cube, and then perform measurements from those new directions, the distribution of the post-measurement location will be a non-symmetric rotation of the previous distribution. That is, it will be a different state.

If we were able to somehow measure location in a single instantaneous measurement, without doing so from any particular direction, we might be able to end up with a post-measurement state that, in the position basis, has the form of a trivariate Gaussian with independent components. That distribution has the rotational symmetries of a sphere, and so can be thought of as coordinate independent. So the question of whether we can make such a measurement is critical.

If the above reasoning is correct then if, as I suspect, an instantaneous measurement of 3D location is not possible, it is not possible to measure location without imposing preferred directions on the system. So there will be a dependence between the final state and the choice of directions from which to measure it.

The above argument about rotational symmetry does not hold if we were able to make exact measurements in each direction. Although we know that is impossible, it might be interesting to investigate whether the state after such an idealised measurement is a function of the axes that were used to measure. For example, is $|\delta^X_1,\delta^Y_1,\delta^Z_0\rangle$ the same as $|\delta^{X'}_{\sqrt 2},\delta^{Y'}_0,\delta^Z_0\rangle$ where $X,Y,Z$ are orthogonal directions and $X',Y'$ are the directions of $X,Y$ rotated 45 degrees around $Z$ so the point that was $(1,1,0)$ in the first coordinate system is $(\sqrt 2,0,0)$ in the second system.

 

[David Olivier]

It's not relevant, because I don't even assume that the “operator” has to have eigenvectors. An ordinary, scalar operator $\mathcal H \to \mathcal H$ will have eigenvectors, but for instance it is well known that the angular momentum “operator” $\vec J$ doesn't have eigenvectors. Only its projection on some specific axis has eigenvectors.

I believe that there is no theoretical obstacle to measuring the three spatial coordinates $(x, y, z)$ of a particle simultaneously, and you believe the contrary. So be it. What I am asking is something else. What should be the expression of such vector (or, for $\vec J$, pseudovector) operators; what kind of objects are they, and what conditions do they satisfy (how do you write that they are self-adjoint, for instance)?

 

[andrewkirk]

It's not relevant, because I don't even assume that the “operator” has to have eigenvectors. An ordinary, scalar operator $\mathcal H \to \mathcal H$ will have eigenvectors, but for instance it is well known that the angular momentum “operator” $\vec J$ doesn't have eigenvectors.

This paragraph doesn't seem to relate to what I wrote. I didn't mention eigenvectors in that latest post, and they play no role in the argument it presents. The argument is based on rotational symmetries of the post-measurement wavefunction in the location basis.

What I am asking is something else. What should be the expression of such vector (or, for $\vec J$, pseudovector) operators; what kind of objects are they, and what conditions do they satisfy (how do you write that they are self-adjoint, for instance)?

The trouble is that thequestion is not well-defined. You ask what 'should' the expression of the vector $(X,Y,Z)$ be. 'Should' according to what principle? We already have an expression of it. It is $(X,Y,Z)$.

In your OP you said you would like to be able to refer to that triple in a way that does not refer to any spatial coordinate system. Implicitly you asked whether that is possible and if so, how it could be done.

My answer is that I am reasonably* confident that it is impossible, and my post above indicates why. You may not agree, but I don't think you can say it is not an answer or not relevant.

I'm very happy to have logical flaws in the above argument pointed out. That will be a learning opportunity for me, and I always relish those. But in the absence of such flaws being identified, it looks like a tough barrier to a coordinate-free representation of the location triple.

*Let's make that 'moderately'. it doesn't do to be too confident of anything in QM.

 

[vanhees71]

Sorry, vanhees71, but it's a mess mainly because almost no one has seriously attempted to answer the original question.

Your answer essentially adds nothing to what I had stated in the original post. I am not asking how to express the three Cartesian components of a vector operator in a particular basis of the 3D space, nor how they transform under $SO(3)$; I am asking how to express them without any choice of a basis in the first place.

It should really not be that difficult to read the question before answering it.

I have no clue what you want then. The components are always defined with respect to a basis. If you refer to Cartesian vector and tensor components then for a physicists these are a operator valued set $\hat{T}_{jk\ldots}$ that transform under rotations $R \in \mathrm{SO}(3)$ via a given unitary representation of this group (or its covering group $\mathrm{SU}(2)$) as
$$\hat{U}(R) \hat{T}_{jkl\ldots} \hat{U}^{\dagger}(R)=R_{ja} R_{kb} R_{lc} \cdots \hat{T}_{abc\ldots}.$$
It's not my fault, if you don't use the usually used terminology.

 

[David Olivier]

This paragraph doesn't seem to relate to what I wrote. I didn't mention eigenvectors in that latest post, and they play no role in the argument it presents.

I think it's basic QM that you can in principle perform a collection of measurements if there are eigenvectors corresponding to the resultant state. This in turn is possible if the observables commute. Since $X$, $Y$ and $Z$ do commute, I think this settles the issue.

You don't see the point in obtaining an expression of quantities such as position, momentum or angular momentum independently from a coordinate system; I do. See below.

I have no clue what you want then. The components are always defined with respect to a basis.

Thanks for informing me that components are defined with respect to a basis.

My question seems quite clear, and I've restated it enough. There is nothing strange or nonstandard in using mathematical objects other than numbers and collections of numbers. There are other things out there, such as vector spaces, tensors, topological spaces and so on. There is nothing unusual in expressing theorems, definitions and physical laws independently from collections of numbers. A vector space, in particular, is not a collection of numbers.

In all domains of physics, there is a natural strive to express laws and concepts independently from any choice of spatial basis. This means expressing them not with coordinates, but with more complex, albeit more abstract, objects.

In special relativity, for instance, the Minkowski vector space can be described as a four-dimensional real vector space $V$ equipped with a bilinear mapping $g: V \times V \to \mathbb R$ such that 1) there exists at least one vector $\vec v$ with $g(\vec t, \vec t) < 0$ and 2) for any two vectors $\vec t, \vec v$ with $g(\vec t, \vec t) < 0$, $\vec v \neq \vec 0$ and $g(\vec t, \vec v) = 0$, we have $g(\vec v, \vec v) > 0$. This is a full description of the metric properties of a Minkowski vector space without the slightest reference to a basis. The whole of special relativity can be expressed in such a way, and there are good reasons to do so.

It is a rather strange and exceptional fact that quantum mechanics are usually expressed with reference to a coordinate system. I was asking if there are known ways to do so. Despite having no answer to that effect, I believe that there probably are, because it is quite normal to want to do so. If you have no interest in the issue, no problem, though.

 

[andrewkirk]

I think it's basic QM that you can in principle perform a collection of measurements if there are eigenvectors corresponding to the resultant state.

Again, this does not address the argument I presented in #29. If anything it adds support to the argument by acknowledging that a collection of measurements is made rather than a single one.

You don't see the point in obtaining an expression of quantities such as position, momentum or angular momentum independently from a coordinate system

I never said that, and it is not true. I greatly prefer coordinate-free presentations, always using them when they are available. That's why I became interested in this thread. But they are not always available, and I'm not going to assume they must be available in every case just because I'd like the world to be that way. If you can disprove my argument above and go on to demonstrate that the post-measurement state of an object will be identical regardless of the directions from which its location is measured, I will be delighted.

 

[Physics Footnotes]

Firstly, let's dispense with the sub-thread started by @andrewkirk concerning the (im)possibility of joint measurements of separate coordinates of a particle in QM. As a bonus, this will segue into the primary thread anyway.

For any (strictly speaking, measurable) region $\Delta \subset \mathbb R^3$ one can construct a projection operator $P_{\Delta}$ with the property that for any pure state $\psi$, $\langle \psi | P \psi \rangle$ represents the probability that the particle is found in the region $\Delta$. In practice, $P_{\Delta}$ represents a particle (location) detector and there are no theoretical limits whatsoever on the shape or size of the region $\Delta$. Any eigenvector of this projector will correspond to a particle localized within $\Delta$, and this makes the use of a 3-dimensional Dirac delta function to describe the limiting case no more controversial than in one dimension.

Note that none of this requires talk of simultaneous measurements from different directions, or anything like that. We just have a single detector which does or doesn't fire. And this is exactly how things work in the real experimental situation too. No coordinates necessary! Of course if you do want to construct this projector with coordinates you can do that too. You can represent a small cuboidal detector, for example, with the projector $P_\Delta := P_{\Delta X}P_{\Delta Y}P_{\Delta Z}$.

And this talk of coordinates provides the segue I referred to above ;-)

In particular, if you want to do QM in a coordinate-free way (and you are absolutely right about the importance of doing this, as well as in your analogy of Minkowski Space), you have to choose the right mathematical language. It turns out that the vector space structure of Euclidean Space $\mathcal E$ is just the wrong way to go about it (even though it raises an interesting mathematical question, which I'm not sure of the answer to). For one thing, $\mathcal E$ just won't capture the non-commutative structure of QM, which is absolutely fundamental. (You still have to accommodate the structure of $\mathcal E$, of course, because that's where the experiment lives; but you don't do that by making the fundamental theory Euclidean.)

So How To Remove Coordinates From The Picture?

The instant you commit to using self-adjoint operators to model observables, you are committing to coordinates, for the following reason. A self-adjoint operator is, at its base, a special kind of mapping called a Projection Valued Measure (PVM) $$P^A:\mathcal B(\mathbb R) \ni \Delta \mapsto P^A(\Delta) \in \mathcal P (\mathcal H)$$It is the projection operators on the right which represent the coordinate-free objects, while the mapping calibrates these questions against a copy of the real number line, based on your choice of coordinate system. The secret to a coordinate-free representation is to strip this arbitrary calibration away, and work purely within the image space, which is a structured set of projection operators.

Specifically, a real observable in QM is a calibration of a coordinate-free Boolean Algebra of projection operators. All the Boolean Algebras of all possible experimental arrangements mesh together in a non-commutative way to form a special type of structure I'll call a Hilbert Lattice, which is more often described as Quantum Logic (although you should not think of the word 'Logic' here as anything more profound that labeling a particular type of algebraic structure.)

Where Does The Euclidean Structure Fit In?

As I mentioned earlier, all experiments live in Euclidean Space (ignoring relativity for now!) and so its structure has to enter the picture somehow. But this is not done directly, by trying to glue quantum objects together to form vectors, but rather by representation theory. That is, the structure of $\mathcal E$ is imposed on the Hilbert Lattice by a representation of its automorphisms (which happen to manifest as unitary operators).

That is why @vanhees71 comment, below, is important (even though it doesn't do what you were trying to do):

As any vector operator the three operators representing the Cartesian components of the position operator transform as a vector (that's why it's called a vector operator in the first place), i.e., for $R \in \mathrm{SO}(3)$ you have a unitary representation $\hat{U}(R):\mathcal{H} \rightarrow \mathcal{H}$ (such a unitary representation must exist since a Galileian invariant QT must admit rotations as symmetry operators). The operators for the Carstesian components of the position operator thus transform under rotations as
$$\hat{U}(R) \hat{\vec{x}} \hat{U}^{\dagger}(R)=R \hat{\vec{x}}.$$

(Although I would have phrased it in terms of coordinate-free projection operators rather than coordinatized self-adjoint operators.)

So if 'coordinate-free' is your real ambition, I would forget about vector operators (except as an interesting mathematical puzzle) and focus on Boolean Algrebras, Lattices, and their Automorphisms (which happen to be induced by unitary operators).

If, on the other hand, you really want to know how to represent operator vectors abstractly, I'm afraid I may have wasted your time with all this.;-)

 

[David Olivier]

Thanks. That's more than I expected!

In the statement that describes a PVM, what is $\mathcal B(\mathbb R)$? The set of measureable subsets of $\mathbb R$? It's not clear to me why we can't have $P^A(\Delta)$ for $\Delta \in \mathcal B(\mathcal E)$.

“and work purely within the image space, which is a structured set of projection operators”. I suppose this is $\mathcal P(\mathcal H)$. How is this defined? Is it the set of linear $p: \mathcal H \to \mathcal H$ such that $p \circ p = p$, or that are self-adjoint with eigenvalues 0 and 1, or something else?

I've found a Hilbert lattice defined in substance as a set of closed linear subspaces of $\mathcal H$, that form a lattice for inclusion. This is a lot for me! Do you have any references that explain some of this?

Is the resultant picture independent also of the origin? And - better still - of the Galilean frame of reference?

 

[Physics Footnotes]

@David Olivier The reason I left out the details is that you had indicated that you didn't want to deal with the subtleties in this thread. So here's a bit more:

• Yes, $\mathcal B (\mathbb R)$ is the Borel Algebra of measurable subsets of $\mathbb R$. You can set up a PVM over any measurable space you like (e.g. $\mathcal B (\mathcal E)$, as long as you define what that means!). I was just pointing out that you get self-adjoint operators when you make that algebra $\mathcal B (\mathbb R^n)$. I suppose leaving $\mathcal E$ abstract, there might be a theorem that gives a representation of a coordinate-free vector operator of sorts, but that is a problem I've never explored. This might be the right way for you to think about this issue though.
• The set $\mathcal P (\mathcal H)$ denotes the orthogonal projection operators on $\mathcal H$. That is, $P^2=P=P^{\dagger}$; the first equality defining projections, while the second guarantees orthogonality. For convenience, I call an orthogonal projection a projector. And yes, the spectrum of any projector is $\{0,1\}$
• Note that the image of a projector is a closed subspace of $\mathcal H$, and it is common practice to work with the lattice of closed subspaces of a Hilbert Space, rather than with $\mathcal P (\mathcal H)$.
• There is no concept of a spatial origin until you impose one. So yes, the group of automorphisms includes translations.

This way of studying QM is very old now. I learned it from the now classic books by Mackey (Mathematical Foundations of QM), Piron (Foundations of Quantum Physics), Jauch (Foundations of Quantum Mechanics), Varadarajan (Geometry of Quantum Mechanics; 2 vols), and others I can't remember off hand. There would surely be some simpler more modern introductory tomes by now, but I'm afraid I'm not familiar with them.

 

[vanhees71]

It is a rather strange and exceptional fact that quantum mechanics are usually expressed with reference to a coordinate system. I was asking if there are known ways to do so. Despite having no answer to that effect, I believe that there probably are, because it is quite normal to want to do so. If you have no interest in the issue, no problem, though.

I try once more to understand what you want. Of course, you can formulate classical mechanics in terms of abstract (Euclidean) vectors, which are simply invariant under rotations.

In QT, observables are represented by self-adjoint operators, which of course you can group into vector components and also operator valued invariant vectors. This is so, because by construction the QT must admit the fundamental space-time symmetry group (at least the part continuously connected to the unit operator).

This is the Galilei group for non-relativistic, i.e., the semidirect product of the temporal and spatial translations, rotations, and Galiei boosts. The symmetries are represented, and this is very important, by socalled unitary ray representations, which can be lifted to a unitary representation of a central extension of the covering group. This results in an 11-dimensional Lie group, where the mass operator is introduced as a central charge of the Lie algebra of the Galilei group and where the rotation group SO(3) is substituted by its coverning group SU(2). The former results in the mass superselection rule in non-relativistic QT the latter in the possibility of half-integer spin.

By definition the operatorvalued vectors are invariant under rotations (i.e., the components transform as given in my previous posting). It's, however important to remember that these vectors are in general not determined, except if all 3 components commute (like for position or momentum). E.g., the vector $\vec{J}$ of angular momentum is almost always never determined but only one component and $\vec{J}^2$.