# The Density Matrix Formalism.

1. Sep 10, 2005

### CarlB

I'm starting to convince myself that the density matrix is a better way of describing a quantum state than a spinor, even in the case of pure states. But it seems like very few of my text books have much to say about density matrices.

Carl

2. Sep 11, 2005

### Juan R.

From a conceptual and general point of view the density matrix formalism is infinitely more powerful and realistic that very outdated and simplistic wave function theory.

From a computational point of view. Density matrix formalism is many times more difficult to solve and this is the reason that wave function is still useful.

In fact, wave packets and the Schrödinguer equation are tipically used fro propagation of density matrices in computational quantum mechanics of condensed matter.

There is a novel formalism that embraces the posibilities of both. It is the TFD. The state is described by a generalized vector |Phy>> that contains full information like the density matrix.

Standard quantum field theory and quantum states in Hilbert space |phy> are recovered in the limit of T--> 0.

I prefer the density matrix formalism in L-space. It is still more powerful and elegant.

Last edited: Sep 11, 2005
3. Sep 12, 2005

### Blackforest

Its for me a very interesting question; may I ask what is the TFD formalism concerning these density matrices? (I need it to go further in my approach). Thanks

4. Sep 13, 2005

### Juan R.

It is so simple like Google search "TFD"

or best still Arxiv Find title or abstract "TFD"

Use also thermal field dynamics.

5. Sep 16, 2005

### Blackforest

Thanks; of course I did it (Google search) and discovered what it means.

6. Sep 16, 2005

### CarlB

It amazes me that my graduate career did not include a decent introduction to density matrices. They were always a derivation, never a fundamental description of nature.

The most amazing part of this is that density matrix formalism eliminates the necessity of having a mathematical representation of an electron rotates by 4 pi when you do a 360. From the density matrix formalism point of view, spinors are just a mathematical convenience. For example:

" Paint each face of a cube a different color and connect each of the eight corners of the cube to the corresponding corners of the room with threads. Now rotate the cube by 360 degrees. The threads are hopelessly tangled up, even though the cube has returned to its original position. Rotation of the cube by another 360 degrees, however, allows one to untangle the threads."
http://www.angelfire.com/stars5/astroinfo/gloss/s.html

I've just spent the most fascinating week working out the details of how one converts calculations from spinors back to density matrix form and back. It certainly was eye opening. I'll write something up in a week or so.

I should mention that I'm looking at the non statistical part of the density matrix formalism. That is, the "pure" states that satisfy:

$$\hat{\rho} \hat{\rho} = \hat{\rho}$$

The mathematicians would call something that satisfies this equation an "idempotent". It turns out that my method of geometrically classifying the elementary particles is based not on the usual arbitrary symmetries applied to wave functions, but instead on the idempotent structure of Clifford algebras. So it naturally leads to a density matrix formalism.

There's another formalism that is centered around the idempotency relation, and that is the Schwinger measurement algebra. And of course quantum mechanics can be written with projection operators.

Carl

7. Sep 17, 2005

### vanesch

Staff Emeritus
Density matrices are strange beasts. If I understand it correctly, the set of all density matrices gives us a complete picture of all we can KNOW about a quantum system. In that respect, they are a bit like the probability distributions over classical phase space. Now "states of our knowledge" of a system are usually not considered to be "states of the system" (in classical parlance, the states of the system are the points in phase space), unless we take a totally epistemological view on nature. For instance, it is hard to expect us for OUR KNOWLEDGE to respect certain symmetries, like the no-FTL propagation. What could it MEAN to say that our KNOWLEDGE about particle B should now be influenced by our KNOWLEDGE of particle A (space-like separated) ?
The STATE of a system could potentially obey such a law (the state of A cannot be influenced by what happens at B, for instance), but our KNOWLEDGE ?
So at first sight, density matrices just describe states of our knowledge while good old quantum states (rays in Hilbert space) describe states of the system.

But then something tricky happens: different mixtures of different states can give rise to identical density matrices (for instance 50% x+ and 50% x- or 50% z+ and 50% z-). Now, in as far as we have then to consider that these are IDENTICAL physical systems, it becomes difficult to maintain that the quantum state |x+> is a state of the system. One should then need to consider the mixture x+/x- from physically different from the mixture z+/z- (even though our knowledge of the systems strictly forbid us to differentiate between both).

So on one hand it is hard to take the density matrix as a fundamental description of the STATE OF THE SYSTEM (but only as a state of our KNOWLEDGE) ; on the other hand, strange things happen like apparently different physical states to be totally indistinguishable.

8. Sep 17, 2005

### Juan R.

An elementary introduction to the subject, and its advantages over usual quantum field theory in

Int. J. Mod. Phys. A 1994, 9(14), 2363-2409.

9. Sep 17, 2005

### Juan R.

Yes, this is a typical error of basic textbooks. In fact, the concept of density matrix cannot be derived from wavefunction, since it is more general.

For mixed states, one work with density matrices. for pure states the concpet of wavefunction arises. In fact, the Schrödinger equation is a special case of the Liouville/von Neumann equation when all probabilities are zero except one.

In fact, physicists' textbooks do not derive density matrices from wavefunctions since they do

1) Assume that pure state is described by |i>

2) Assume that real quantum state is described by a population of different |i> with probabilities Pi, then we define

rho = Sum |i> Pi <i|

Obiously the point (2) is outside of the framework of QM since Pi cannot be computed from any wavefunction.

However, one can obtain wavefunction theory from the general rho.

Since that wavefunction theory, and all standard quantum mechanics arise from the limit Pi = 0 for all i =/= j

Then the state is rho = |j><j| and this is equivalent to work directly with |j> or with <j| in usual QM. In fact, the Schrödinguer equation is derived, the theorem for computation of observables via |j> is computed, etc.

Precisely i am working in a theorem where show that usual coarse grained interpretation of density matrices is completely wrong without mathematical basis. I wait publish it this year.

Last edited: Sep 18, 2005
10. Sep 18, 2005

### CarlB

Projection operators.

To give a preview of what I am working on with idempotents and elementary particles:

Let us associate elementary particle states with idempotents. That is, use the pure density matrices to represent the particles. Then these idempotents satisfy:

$$\iota_A \; \iota_A = \iota_A$$

Continuing to follow the Schwinger measurement algebra, we will assume that the idempotents, $$\iota_\chi$$ are "primitive", that is, that each cannot be written as the sum of nontrivial idempotents. Furthermore, we will assume that a sum over a complete set of idempotents gives unity:

$$\sum_\chi \;\iota_\chi = 1$$

As an example of this sort of thing, use the Dirac algebra. Then four idempotents would correspond to the electron and positron with spins up or down (or right/left) as measured in the z direction.

The next thing that comes to mind is to consider the electrons and positrons with spin up or down as measured in some direction other than z. These would be represented by four primitive idempotents, but they would be distinct from the above four idempotents.

We might as well go to the case of arbitrary spin directions, so let $$\hat{u}$$ and $$\hat{v}$$ be two directions that are not parallel or antiparallel. Let $$\theta$$ be the angle between u and v. We now have 8 different density matrices, they represent the electron and positron with spins up or down in the u or v directions.

Here's the notation we will use:

Electron spin up in u direction: $$e_{u+}$$
Positron spin down in v direction: $$\bar{e}_{v-}$$

Now consider the possible products of these matrices.

First, the idempotents associated with u (or v) will be orthonormal, sort of. More particularly, the identical idempotents will square to the same thing, and distinct idempotents (but with the same orientation) will multiply to zero (annihilate). For example:

$$e_{u+}\;e_{u+} = e_{u+}$$
$$e_{u+}\;e_{u-} = 0$$
$$e_{u+}\;\bar{e}_{u-} = 0$$
$$e_{u+}\;\bar{e}_{u-} = 0$$

Second, electrons and positrons, even with distinct orientations, will annihilate:

$$e_{u+} \; \bar{e}_{v\pm} = 0$$
etc.

But electrons (or positrons) with distinct orientations will not annihilate, even if they have different spin:

$$e_{u+}\;e_{v+} \neq 0$$

Now consider the vector space generated by a basis consisting of all possible multiples of the 8 matrices we have described. How big of a vector space is it?

It turns out that a complete basis set for the products of these 8 matrices consist of the 24 matrices:

$$e_{u+}$$
$$e_{u-}$$
$$\bar{e}_{u+}$$
$$\bar{e}_{u-}$$
$$e_{v+}$$
$$e_{v-}$$
$$\bar{e}_{v+}$$
$$\bar{e}_{v-}$$

$$e_{u+}\;e_{v+}$$
$$e_{u+}\;e_{v-}$$
...
$$e_{v-}\;e_{u-}$$
$$\bar{e}_{u+}\;\bar{e}_{v+}$$
...
$$\bar{e}_{v-}\;\bar{e}_{u-}$$

That is, any possible product of these eight matrices will reduce to a real multiple of one of the 24 above. And it turns out that the multiples, that is the ratio of the product to the basis set listed above, gives the branching ratio (or trace) computed for a transition through the given sequence of measurement algebra (generalized Stern-Gerlach apparata, i.e. Stern Gerlach apparata that pass only one particle type and spin and annihilate all other particles). For example:

$$e_{u+}\;e_{v+}\;e_{u+} = ((1+\cos(\theta))/2)^2 e_{u+}$$

Just as we would expect to see a reduction in intensity by $$(1+\cos(\theta))/2$$ for each of the two transitions through Stern-Gerlach apparata.

A more interesting product:

$$e_{u+}\;e_{v-}\;e_{u-}\;e_{v+} = \alpha e_{u+}\;e_{v+}$$

The above product is equivalent to the action of four consecutive Stern-Gerlach apparata. But the overall effect is equivalent to the product of two Stern-Gerlach apparata with a reduction in the overall intensity. Have I made the explanation clear enough that anyone can calculate $$\alpha$$ , the reduction factor?

Now the thing that is interesting is that so long as we restrict ourselves to only two different directions, we can get by with real multiples. But when we go to three different directions, perhaps we might call them R, G and B, then the reduction factor can become complex.

Carl

11. Sep 18, 2005

### CarlB

For example, consider the Pauli case and three Stern-Gerlach apparata with orientations of x, y and then z. The density matrix for the overall experiment is given by the product of the three projection operators:

$$P_x\;P_y\;P_z$$
$$=(1+\sigma_1)(1+\sigma_2)(1+\sigma_3)/8$$
$$= (1+\sigma_1+\sigma_2+\sigma_3 +i\sigma_3 +i\sigma_1 -i\sigma_2 +i\sigma_3\sigma_3)/8$$
$$= \frac{1+i}{2} (1+\sigma_1)(1+\sigma_3)/4$$
$$= \frac{1+i}{2} P_x \; P_z$$

In other words, we can again write the product of the idempotent matrices as a product where we include only the first and last matrices. All the idempotents that show up in the middle get turned into just a complex constant that factors out.

The same calculation obtains if one uses the gamma matrices, but the factor may be a bit more complicated than just a complex number. Depending on how one embeds SU(2) in the Dirac algebra, one can end up with several different possibilities for the "i". For example, one might wish instead to use an embedding which gave $$\gamma_0$$ instead of i. If this sort of thing happens, then it turns out that whatever thing we end up that is more complicated than i will nevertheless commute with our surrounding idempotents.

In other words, it really is the case that a sequence of Stern-Gerlach apparata can be reduced to just the first, last, and a phase multiplication, and this is the case even when we assume Stern-Gerlach apparata that can separate arbitrary isospin. The freedom to embed SU(2) in several different ways corresponds to a freedom to assume Stern-Gerlach apparata that measure isospin, spin and or various combinations.

Now the reason that this is particularly interesting is that it shows a place where one can extract complex numbers from QM in a manner that is not simply a result of our use of spinors. That is, we get complex numbers even within the density matrix formalism, and not only that, but they show up in the density matrix formalism without any need to deal with wave functions that depend on position. The reason this is important is that it gives us a way of distinguishing between the complex phases that are just phases and that go away in a density matrix formulation, and the complex phases that allow quantum waves to intefere with themselves.

I should mention that the whole concept of the Schwinger measurement algebra seems to be to associate the elementary particles with our representations of the measurements that define them. That this would produce a formalism that has less garbage (i.e. nonphysical phase degrees of freedom) than the spinors is interesting to me.

Carl

Last edited: Sep 18, 2005
12. Sep 20, 2005

### CarlB

A more general result:

Let $$u,v,w$$ be unit vectors, and $$P_u,P_v,P_w$$ be the corresponding spin operators defined by:

$$P_u = \frac{1}{2}(1+\sigma_xu_x+\sigma_yu_y+\sigma_zu_z)$$

etc. Then one has:

$$\left(1 + u\cdot w +(u+w+i(w\times u))\cdot v) \right)P_uP_w = 2(1+u\cdot w)P_uP_vP_w$$

If $$u$$ and $$w$$ are antiparallel, then the equality reduces to $$0=0$$. Otherwise, it can be used to simplify products of projection operators.

This has some utility when one is working with the density matrix formalism in that one represents a spin state with a projection operator.

If one wishes to make a Zitterbewegung type model of elementary particles, one must make somewhere the assumption that a particle with one spin state spontaneously changes into another and then back. If one assumes that the particles are composite, then the natural extension to the Zitterbewegung theory is to allow for the possibility that the subparticles switch from one sort (i.e. spin direction and or particle type) to another. One can use the above to calculate the matrix that shows how the switches occur, and from that matrix, one can derive the composite mixtures that are preserved by the spontaneous transformations.

It should be noted that when $$u=w$$, the above calculation shows that $$P_uP_vP_w$$ is a positive real <= 1 multiple of $$P_uP_w = P_uP_u = P_u$$, but when $$u \neq w$$, the multiple can be complex. Switching $$u$$ with $$w$$ causes a complex conjugation on the ratio, so a matrix made of these sorts of things will be Hermitian.

My motivation for going through all this is that I found a Hermitian matrix that has a form suitable to be made up of products of spin projection operators which happens to have eigenvalues equal to the square roots of the masses of the electron, muon and tau. The matrix is shown on page 8 of this paper by Alejandro Rivera and Andre Gsponer:
http://arxiv.org/abs/hep-ph/0505220

The implication here is that the leptons are composites. The subparticles undergo spontaneous spin changes that create a Zitterbewegung-like composite particle.

Carl

Last edited: Sep 20, 2005
13. Oct 2, 2005

### Tom Mattson

Staff Emeritus
I'm sorry that this one got by us, but I have just deleted all references to Juan's personal website and the discussion that followed from it. I would like to state with emphasis that we host discussions of personal theories at one and only one place at Physics Forums, and that is in the Independent Research Forum.

Last edited: Oct 2, 2005
14. Oct 3, 2005

### solidspin

We use density matrices all the time w/r/t our pulse sequences in solid-state NMR. They're extremely powerful from a practical pov, b/z the off-diagonal terms represent coherences w/ very useful physical meaning.

A reasonable book w/ some very good practical application is called Spin Dynamics, by Malcolm Levitt. Of course, this is about NMR, but you will quickly see that manipulation of large nxn matrices, after some manipulation of sandwich operators, yields some great results w/o too much headscratching.

15. Oct 4, 2005

### CarlB

Dirac's transformation theory

An interesting comment on the foundations of quantum theory from a web published book:

From c-Numbers to q-Numbers
The Classical Analogy in the History of Quantum Theory
Olivier Darrigol
(University of California Press, 1993)

Chapter XIII:

There is one feature of Dirac's original transformation theory that is likely to surprise the modern quantum physicist: the notion of state vector is completely absent. It was in fact introduced later by Weyl and von Neumann, and subsequently adopted by Dirac himself. In 1939 Dirac even split his original transformation symbol (x '/a ') into two pieces $\langle \xi' |$ and $|\alpha' \rangle$ , the "bra" and the "ket" vectors.[92] The mathematical superiority of the introduction of state vectors is obvious, since it allows—albeit not without difficulty—an explicit construction of mathematical entities (rigged Hilbert spaces) that justify Dirac's symbolic manipulations. There was also a more physical advantage to the notion of state vector: it placed the superposition principle in the foreground, which pleased Bohr, who set wave-particle duality at the core of complementarity. Perhaps modern-day interpreters of quantum mechanics should nevertheless remember that there exists a formulation of quantum mechanics without state vectors, and with transition amplitudes (transformations) only.
http://upstage.cdlib.org/xtf/view?docId=ft4t1nb2gv&brand=eschol

What I find interesting about the above is that Dirac's original formulation avoided the problem of states having unphysical phases. The primary advantage for the bra ket formulation is that it allows linear superposition of states. But this advantage is a calculational one only. One cannot physically take linear superposition of states. For example, three times a wave state is not the correct wave state for three particles.

The choice between the two formulations is to either have simple math with unphysical degrees of freedom, or alternatively to have complicated math without the unphysical degrees of freedom. My guess is that the most advantageous choice is to use the density matrix formalism for the theory, but to do practice with the state vector formalism. For that purpose, we must understand how to convert back and forth between the formalisms and this is what I've been working on recently.

Carl

16. Nov 3, 2005

### Hans de Vries

This is interesting stuff Carl. I'm looking for a book which handles this in
some detail. Do you have any suggestions?

This one seems fine: Karl Blum: Density Matrix Theory and Applications
(Physics of Atoms and Molecules)
, but it's $157.00 :^( http://print.google.com/print?hl=en...=PA21&pg=PA21&sig=mTl9FBvg9JkShMxuPdB87pX6SyY This one is$75: From Classical to Quantum Mechanics : An Introduction
to the Formalism, Foundations and Applications

It's a decent scientific site, but:

To see how the wires go: Imagine a pyramid on top of the cube (pointing up)
and a pyramid under the cube (pointing down). The edges are the wires. the
threads continue from the tops of the pyramids to the corners of the room.
They can turn what they want but the threads never get untangled :^)

Regards, Hans

17. Nov 4, 2005

### CarlB

Hans, I take your comment as a high compliment. I am the source for this. Of course I can't realistically claim to be the originator because it is a rather obvious method of looking at a subject that has been studied extensively for many decades. It's from the paper I'm writing on the subject of how one derives masses for the fermions. Look on page 10 and following of this link:
http://www.brannenworks.com/long_PANIC_Not_Complete_.pdf

The above link also gives a beautiful relationship between the areas carved out by a series of Stern-Gerlach experiments and the phase change associated (on page 12).

The whole reason for working this out is to explain that factor of $$e^{i\delta}$$ in that matrix form for the lepton mass matrix over on the lepton mass thread. Basically, the value of $$\delta$$, along with the Koide relationship, defines the masses of the leptons. From the point of view of "binon" theory (which theory I am now revising so that it is compatible with the gauge symmetry breaking stuff I did for Santa Fe), delta is defined by the angle between the three binons that make up a lepton and the lepton's axis itself. That is, the subtended area induces a complex phase according to the formula. This is different from the $$\cos(\theta_B) = 1/3$$ I was assuming earlier, but 1 meter forward, 90 centimeters backward is good progress.

I looked in some of the density matrix formalism books at the local library, but none of them were really devoted to examining the foundations of QM. The authors treated it as if it were just another way of making convenient calculations.

The first reference you gave seemed to be working on products of Pauli spin matrices. That's not what I'm working on here. I'm multiplying products of Pauli spin projection operators. The reason for this is that this is what one gets when one puts multiple Stern-Gerlach apparata together, and if one is to define a particle according to what experiment produces it, then products of Stern-Gerlach apparata show that one must include an arbitrary phase.

Carl

Last edited: Nov 4, 2005