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JMP> Vol.9 No.4, March 2018
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Algebra of Classical and Quantum Binary Measurements

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DOI: 10.4236/jmp.2018.94044    347 Downloads   1,293 Views   Citations
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Carl A. Brannen


Retired, Henderson, NV, USA.


The simplest measurements in physics are binary; that is, they have only two possible results. An example is a beam splitter. One can take the output of a beam splitter and use it as the input of another beam splitter. The compound measurement is described by the product of the Hermitian matrices that describe the beam splitters. In the classical case, the Hermitian matrices commute (are diagonal) and the measurements can be taken in any order. The general quantum situation was described by Julian Schwinger with what is now known as “Schwinger’s Measurement Algebra”. We simplify his results by restriction to binary measurements and extend it to include classical as well as imperfect and thermal beam splitters. We use elementary methods to introduce advanced subjects such as geometric phase, Berry-Pancharatnam phase, superselection sectors, symmetries and applications to the identities of the Standard Model fermions.


Quantum Mechanics, Schwinger Measurement Algebra, Quantum Thermodynamics

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Brannen, C. (2018) Algebra of Classical and Quantum Binary Measurements. Journal of Modern Physics, 9, 628-650. doi: 10.4236/jmp.2018.94044.

1. Introduction

In 1955, Julian Schwinger began work on the foundations of quantum field theory while employed at Harvard. The result was what is now known as “Schwinger’s Measurement Algebra”. The algebra was described in four of his 1959-1960 papers: [1] [2] [3] [4] . Schwinger used his algebra to teach introductory quantum mechanics. He joined the faculty at the University of California, Los Angeles in 1972 and his lecture notes from there resulted in two textbooks [5] [6] that many of his students then used when they taught introductory quantum mechanics. These textbooks cover the usual subjects in a standard quantum mechanics class. Where they are distinct is in their introduction to the subject, the algebra that this paper expands.

Schwinger’s first paper “The Algebra of Microscopic Measurements” [1] describes his measurement algebra. His measurements can be thought of as beam splitters where one is concerned with only one of the exits. The algebra relates to how one models a complex beam splitter that consists of a series of beam splitters connected together by arranging for the output of one beam splitter to be used as the input of the next. The reader of Schwinger’s paper will note that while his notation is different, the properties of the elements of his algebra are similar to those of mixed density matrices. We will use density matrix notation in this paper. We expand Schwinger’s results to include the classical situation as well as imperfect and thermal beam splitters. To simplify the discussion, we will mostly consider binary measurements.

There are two reasons for reading this paper. The first is that Schwinger’s beam splitter model provides the most direct method of passing from the classical to the quantum domain. Since Schwinger’s method is completely general for quantum mechanics and quantum field theory, this provides an immediate connection between the classical and quantum situations and may provide an improved understanding of the foundations of quantum mechanics for students. In addition, binary measurements are arguably the easiest introduction to quantum physics and we introduce subjects that usually require much more preparation such as geometric phase, Berry-Pancharatnam phase and quantum statistics. The second reason is that some problems in quantum mechanics are far easier to understand in one formulation than in the others [7] . Schwinger’s formulation is particularly useful in putting the symmetries of quantum mechanics on an algebraic basis. We demonstrate this by introducing a Schwinger algebra model of the elementary fermions.

1.1. Beam Splitters

The physical apparatus we’re considering is a beam splitter. The beam splitter has a single entrance and two exits. Entering particles must take one of the two exits. Suppose that the particles are all identical of type “1” and that they do not influence one another and that they arrive at a beam splitter with a rate | A 1 | 2 . We use | A 1 | 2 here (instead of A 1 ) for our rates in order to match the notation in the physics literature for beam splitters described with quantum mechanics. Our A 1 is the square root of a rate, so its units are particles / second . In general, A 1 may be complex.

Suppose that a particle has a probability p 1 of exiting the upper exit so that the probability of exiting the lower exit is 1 p 1 . Then the rate of particles in the upper exit beam is | B 1 | 2 = p 1 | A 1 | 2 and the rate at the other exit is ( 1 p 1 ) | A 1 | 2 = | B 1 | 2 | A 1 | 2 . See Figure 1.

If we know the input rate and the upper exit output rate, then we can obtain the lower exit output rate by subtraction. Accordingly, for the remainder of this

Figure 1. Trivial beam splitter splits a beam of type 1 particles arriving at a rate | A 1 | 2 . Particles take the upper exit with probability p 1 so the output rate for the upper exit is | B 1 | 2 = p 1 | A 1 | 2 and the output rate for the lower exit is | A 1 | 2 | B 1 | 2 = ( 1 p 1 ) | A 1 | 2 . We’ve written the equations in terms of amplitudes A j rather than rates | A j | 2 in order to match the standard quantum beam splitter notation.

paper, we will ignore the lower exit and concentrate only on the upper exit. For clarity, our drawings will continue to include the lower exit but we will not write formulas for it.

1.2. Outline

Schwinger noted that what makes quantum measurements different from classical is that a quantum measurement of one property (say spin in the +z direction) can disturb the system so that the results of a previous quantum measurement (of spin in another direction), no longer applies. Schwinger considered compound beam splitters obtained by putting the output of a first beam splitter into the input of a second beam splitter. He showed that these experiments can be represented by Hermitian matrices and that a compound beam splitter is represented by the product of its constituent beam splitter matrices.

In general, Hermitian matrices do not commute. For the Schwinger measurement algebra this means that changing the order of measurements can have a physical effect. If in addition to being Hermitian the matrices are also diagonal, then they will commute and the measurement order does not matter. In Section 2, we consider classical beam splitter experiments with imperfections or at temperature. We show that these experiments can be modeled with diagonal Hermitian matrices, that is, with real diagonal matrices.

Section 3 continues the analysis to the quantum case by allowing for nonzero off diagonal entries in the matrices. For simplicity, we specialize to spin-1/2 Stern-Gerlach experiments. We find that the quantum situation is a natural extension of the classical situation. We show why quantum mechanics uses complex numbers and amplitudes instead of probabilities, and we introduce the ideas of geometric phase, Berry-Pancharatnam phase, superselection sectors, quantum symmetries and quantum statistics.

As a unique formulation of quantum mechanics, Schwinger’s measurement algebra can be expected to provide unique applications to Nature. Our analysis of superselection sectors concludes that their algebras are block diagonal in form. This suggests that we reverse the process. We can start with an algebra and from it derive the particle content. We include Section 4 as a speculation on the nature of the Standard Model fermions and dark matter. The paper concludes with a conclusion and acknowledgements.

2. Classical Beam Splitters

The two outputs of the beam splitter of Figure 1 have intensities | B 1 | 2 and | A 1 | 2 | B 1 | 2 . These intensities add up to | A 1 | 2 , the intensity of the original beam. If we combine the two exits back together we obtain an exit beam with intensity | A 1 | 2 and expect that this beam will be indistinguishable from the original beam.

The beam splitter of Figure 1 is described with only a single number, the probability of a particle leaving the top exit, p 1 . Suppose we have another beam splitter, this one with the corresponding probability q 1 . Assuming independence, a particle will make it through both beam splitter’s top exits with probability q 1 p 1 . See Figure 2. This process can be continued with any number of consecutive beam splitters. If we have n beam splitters with probability p 1 , the probability of a particle making it through all of them is ( p 1 ) n .

2.1. Classical Thermal Beam Splitters

To introduce thermal effects, let’s suppose that a trivial beam splitter has its two probabilities p 1 and 1 p 1 , depend on temperature T according to a positive energy difference Δ E 1 and temperature T via the Boltzmann factor

p 1 e x p ( Δ E 1 k b T ) , 1 p 1 e x p ( + Δ E 1 k b T ) . (1)

The proportionality can be determined by the requirement that p 1 and 1 p 1 add to 1. So we divide by the sum of the right hand sides to get the

Figure 2. Two trivial beam splitters connected together. Particles of type 1 arrive at rate | A 1 | 2 . The beam splitters have probability p 1 and q 1 so the overall probability is the product p 1 q 1 and the output beam has a rate of | B 1 | 2 = p 1 q 1 | A 1 | 2 .


p 1 = exp ( Δ E 1 k b T ) / ( exp ( Δ E 1 k b T ) + exp ( + Δ E 1 k b T ) ) , 1 p 1 = exp ( + Δ E 1 k b T ) / ( exp ( Δ E 1 k b T ) + exp ( + Δ E 1 k b T ) ) . (2)

In the high temperature limit k b T Δ E 1 , the probabilities approach p 1 = 1 p 1 = 1 / 2 .

At the low temperature limit, the probabilities go to p 1 = 1 , p 0 = 0 and near zero, they are approximately

p 1 e x p ( 2 Δ E 1 k b T ) , p 2 1 e x p ( 2 Δ E 1 k b T ) . (3)

In this cold limit, putting n beam splitters together by taking the exit ports to the input port of the next beam splitter will give a final upper exit port probability of p 1 n and this corresponds to decreasing the temperature from T to T/n:

( p 1 ) n ( exp ( 2 Δ E 1 k b T ) ) n = exp ( 2 Δ E 1 k b T / n ) , (4)

Thus connecting identical thermal beam splitters has the effect of reducing the temperature. This temperature effect will also work in the quantum situation.

2.2. Classical Two Particle Beam Splitters

The reader may be relieved to read that now we consider beam splitters that act on a particle beam with two kinds of particles, type 1 and type 2. The two particles arrive with rates | A 1 | 2 and | A 2 | 2 and the probabilities that they leave via the upper exits are p 1 and p 2 . If the output of the beam splitter is sent to another identical beam splitter, the probabilities will square so that

| B 1 | 2 = p 1 p 1 | A 1 | 2 = ( p 1 ) 2 | A 1 | 2 , | B 2 | 2 = p 2 p 2 | A 2 | 2 = ( p 2 ) 2 | A 2 | 2 . (5)

We can rewrite the above equations in 2 × 2 diagonal matrix form:

( | B 1 | 2 0 0 | B 2 | 2 ) = ( p 1 0 0 p 2 ) ( p 1 0 0 p 2 ) ( | A 1 | 2 0 0 | A 2 | 2 ) (6)

See Figure 3.

The reader may notice that we could have put the | B j | 2 and | A j | 2 numbers into vectors instead of matrices. We’re doing it with matrices so that our presentation will be compatible with the density matrices of quantum mechanics. For arguments supporting density matrices as a superior way of representing quantum states see [8] .

Figure 3. Two identical, classical beam splitters connected together. Particles of type 1 and 2 arrive mixed together in a single beam with rates | A 1 | 2 and | A 2 | 2 . The beam splitters have probability p 1 and p 2 for the two types so their overall probabilities are ( p 1 ) 2 and ( p 2 ) 2 . This can be expressed with matrix multiplication.

With the probabilities for the two particle types being p 1 and p 2 , the corresponding probabilities for the lower exit will be 1 p 1 and 1 p 2 . This is represented by a matrix with those numbers on the diagonal. So if we recombine the lower and upper exit ports the resulting experiment will be represented by the sum of the two matrices which is the unit matrix:

( p 1 0

Any two points on the surface of the sphere define a great circle route unless those two points are on opposing ends of a diameter (antipodal). For a unit vector u, the points u and -u are on the opposite ends of a diameter and we have

ρ u ρ u = 0. (34)

That is, the projection operators for opposite spin measurements annihilate each other and there is no unique great circle route to make the transition adiabatic.

The usual introduction to spin-1/2 uses raising and lowering operators. For the spin-1/2 case, these are operators that negate the direction of a quantum state (amplitude), for example the raising operator for spin-1/2 takes the spin-down state to a spin-up state. More general raising and lowering operators change a state from one spin state to an orthogonal spin state and again the projection operators annihilate.

The absence of a great circle route between annihilating spin projection operators means that there is no natural phase that can be assigned for such an operator. For example, the raising operator for spin-1/2 can be written as:

ρ + ( θ ) = ( 0 exp ( i θ ) 0 0 ) (35)

where θ is any angle. In the standard presentation of raising operators, θ is chosen as zero. We can obtain the general case by making a choice for the intermediate measurement along the equator of the sphere at the point u ( θ ) = ( cos ( θ ) , sin ( θ ) , 0 ) so that

ρ + u ( θ ) = ( 1 + c o s ( θ ) σ x + s i n ( θ ) σ y ) / 2 . (36)

Then the arbitrary phased raising operator can be written as a product of projection operators:

( 0 exp ( i θ ) 0 0 ) = 2 ρ + z ρ + u ( θ ) ρ z (37)

where the factor of 2 is included due to the 1 / 2 loss in amplitude in the two transitions. The standard raising operator is obtained by choosing the intermediate measurement as ρ + x = ( 1 + σ x ) / 2 .

3.5. Quantum Temperatures

Temperatures work the same in the quantum situation as they do in the classical. Using the classical numbers from Equation (1), a quantum beam splitter acting on a beam with only a single particle type will be identical to the classical case. To make it more interesting we can consider such a device acting on a beam with two particle types. Consider a measurement of spin-1/2 in the +z direction. A perfect measurement (temperature zero) is given by ( 1 + σ z ) / 2 . At finite temperatures some of the spin-up particles end up in the lower exit and some of the spin-down particles take the upper exit. Using + Δ E for the first particle type and Δ E for the second we have a matrix for the upper exit port:

ρ ( T ) = ( exp ( + Δ E k b T ) 0 0 exp ( Δ E k b T ) ) / ( exp ( + Δ E k b T ) + exp ( Δ E k b T ) ) (38)

In the low temperature limit, and choosing to keep the sum of the probabilities 1, these become

ρ ( T ) ( 1 e x p ( 2 Δ E k b T ) 0 0 e x p ( 2 Δ E k b T ) ) , (39)

similar to the classical result in Equation (4).

We can connect n of these experiments together so that the upper exit of one beam splitter goes to the entrance of the next; the composite experiment is modeled by the n power of the matrix ρ ( T ) :

( ρ ( T ) ) n ( 1 e x p ( 2 Δ E k b T / n ) 0 0 e x p ( 2 Δ E k b T / n ) ) . (40)

The above is diagonal, as would be obtained for a classical thermal two particle beam splitter. To get off diagonal elements we need to consider spin in directions other than ± z . Our calculations have been for spin-1/2 in the +z direction but that is not a special direction so we can generalize to the +u direction. To do that, we need to rewrite the above in the Pauli spin matrix basis. We find

( ρ ( T ) ) n ( 1 ^ + ( 1 e x p ( 2 Δ E k b T / n ) ) σ + u ) / 2 . (41)

where σ z has been replaced with σ + u to give the general spin-1/2 case.

Squaring a low temperature matrix ρ ( T ) approximately gives the matrix for an even lower temperature ρ ( T / 2 ) . For general temperatures, the squaring does not necessarily divide the temperature by two but (other than the high temperature limit) it does reduce the temperature. The high temperature limit has probabilities 1/2:

ρ ( T Δ E / k B ) ( 1 / 2 0 0 1 / 2 ) . (42)

Since the matrices for the upper and lower exits must sum to 1 ^ , the lower exit matrix must be identical to the above. Squaring the above matrix gives a matrix with 1/4s on the diagonal instead of 1/2s but the matrix must still correspond to the same high temperature limit so we need to multiply the matrix by 2. This “renormalization” was (approximately) unnecessary at very low temperatures. At higher temperatures, one can maintain the trace as one by dividing the matrix by the trace after squaring. The trace of the above matrix is 1/2 so dividing by 1/2 will renormalize it. Density matrices are beyond the scope of this paper however we note that requiring our matrices to have trace 1 makes them mathematically identical to “mixed density matrices” and when density matrices are used in statistical physics, squaring and renormalization is a method used to reduce their temperature [10] .

3.6. Superselection Sectors

Suppose we have a beam of spin-up electrons and we split it with a beam splitter measuring spin-1/2 in the u = ( u x , u y , u z ) direction. The entering beam is represented by a ket that is pure spin-up:

| + z = ( 1 0 ) , (43)

and the matrix representing the measurement is

ρ + u = ( 1 ^ + σ u ) / 2 = 1 2 ( 1 + u z u x i u y u x + i u y 1 u z ) . (44)

Performing the matrix multiplication ρ + u | + z , the exiting beam is represented by a ket for spin-1/2 in the +u direction:

| + u = 1 2 ( 1 + u z u x + i u y ) = 1 + u z 2 | + z + u x + i u y 2 | z . (45)

The above shows that it is physically possible to begin with a beam of pure spin-up and from it make a beam that is a “linear superposition” of spin-up and spin-down. This is accomplished by arranging for the matrix ρ + u to have non zero off diagonal elements. That is, we will get a linear superposition provided u x or u y is non zero.

It’s natural for students to imagine that it’s possible to create a superposition of any quantum states, given that physicists talk about superposition of states consisting of a live and dead cat. But in fact, not all quantum superpositions can be created; superpositions are limited by “superselection sectors” and this is what we will discuss in this subsection. Superselection sectors can be attributed to two subjects beyond the scope of this paper, symmetry and decoherence so we will provide only an outline of the ideas. The subject is explained in [11] ; we will explore the subject from the point of view of binary measurements.

Our starting point is to assume, as Julian Schwinger did, that given a “complete set of commuting observables” to define a quantum state, it is possible to define a beam splitter whose upper exit passes only that particular state, for example, spin-up. The spin-1/2 symmetry defines states for all the possible directions u so we can use intermediate states to obtain a superposition between spin-up and spin-down as was illustrated in Equation (45).

We will now consider a binary measurement on a particle beam that contains electrons and neutrinos. There are no intermediate quantum states between the electron and neutrino so there is no way for us to use our spin-1/2 example to convert an electron beam into a beam that is a superposition of electron and neutrino. This is the most common case for “internal” symmetries, however, the particle “generations” are an exception and in fact, the weak force converts an electron into a linear superposition of neutrinos from different generations.

Since it’s not possible to create linear superpositions of electrons and neutrinos, it is only possible to make statistical mixtures of these particles. This is the same case as the classical beams we considered in the previous section. Classical beams are represented by matrices of particle rates that are diagonal as in Equation (6). Before we split the particle rates into bras and kets, we represented spin-1/2 particles with 2 × 2 Hermitian matrices.

By reconsidering our splitting we can create a 4 × 4 matrix that represents a statistical mixture of electrons and neutrinos, each of which is a quantum superposition of spin-up and spin-down. Suppose our beam is 40% electrons with spin-1/2 in the +u direction and 60% neutrinos with spin-1/2 in the +v direction. Then the matrix representation is:

ρ ( T ) = 0.4 | + u , e + u , e | + 0.6 | + v , ν + v , ν | = 0.4 ρ + u , e + 0.6 ρ + v , ν = ( 0.2 + 0.2 u z 0.2 u x 0.2 i u y 0.2 u x + 0.2 i u y 0.2 0.2 u z 0.3 + 0.3 v z 0.3 v x 0.3 i v y 0.3 v x + 0.3 i v y 0.3 0.3 v z ) (46)

where T is some arbitrary temperature and, for the sake of clarity, we’ve left the forbidden matrix entries blank instead of zero. This matrix is in the form of a mixed density matrix subject to the requirement that its bras and kets not cross the superselection sector boundary. That is, the kets are split into two halves and only one of the halves can be non zero. If it is the top half, then the ket is an electron ket while the bottom half defines a neutrino ket.

We’ve just shown that the particle rate matrix for beams that include more than one superselection sector must be in block diagonal form. The splitting of two superselection sectors is a classical beam splitter and this can always be done. After reducing a beam to a single superselection sector we can do quantum measurements on it with the methods discussed above. We can do this to both the particle types and then reassemble the two beams into one. This way we can design a beam splitter whose output stream will have the properties of 40% electron and 60% neutrino beam discussed above, at least given an input stream with both electrons and neutrinos.

We can also consider thermal measurements of beams composed of particles from two different superselection sectors. As before with Equation (42), the high temperature limit for each superselection sector will be a multiple of the unit matrix for that sector. If we balance the sectors to have the same multiple, the high temperature limit will be a multiple of unity. Setting the trace to be 1, the high temperature limit for the electron/neutrino block diagonal matrix will be:

1 4 ( 1 0 0 1 1 0 0 1 ) , (47)

where again we’ve left the forbidden entries blank rather than zero.

We can reduce the temperature of the above matrix ρ ( T ) in Equation (46) by squaring and renormalizing to keep the trace 1. This is easily done in the form 0.4 ρ + u , e + 0.6 ρ + v , ν as ρ + u , e and ρ + v , ν are idempotent, annihilate each other and have unit trace. We find:

ρ ( T ) n / t r ( ρ ( T ) n ) = ( ( 0.4 ) n ρ + u , e + ( 0.6 ) n ρ + v , ν ) / ( ( 0.4 ) n + ( 0.6 ) n ) (48)

In the limit as n the ( 0.6 ) n terms dominate giving a limit

ρ ( 0 ) = ρ + v , ν . (49)

It should be clear that this is a general attribute of quantum beam splitters that cross superselection sectors. That is, their T = 0 limit falls into a single sector.

If we are able to make calculations in an algebra and want to know what particles it contains this implies an algorithm. We begin with the high temperature limit of Equation (47) and add a small Hermitian modification to it. Then repeatedly square and renormalize the trace to unity. It will approach the primitive idempotent for a particle.

For our example of the electron/neutrino, the operator for electric charge is zero in the neutrino part and 1 ^ = 1 1 ^ in the electron part:

Q = ( 1 0 0 1 0 0 0 0 ) . (50)

This matrix commutes with any element of the measurement algebra for the electron/neutrino beam. In fact, the definition of a charge that creates superselection sectors is that the symmetry must commute with every possible measurement (observable).

Given an algebra, we can define the possible charges that define its superselection sectors. Each diagonal block can take a different charge. For the electron/neutrino algebra, the possible charges are:

Q ( q e , q ν ) = ( q e 0 0 q e q ν 0 0 q ν ) . (51)

where q e and q ν are the charges for the electron and neutrino blocks. This observation is trivial for block diagonal algebras but becomes interesting when the algebra is defined more subtly.

4. Standard Model/Dark Matter

The Standard Model of elementary particles is built around symmetries. This is a natural consequence of the experiments; humans look for patterns in the experimental results and the nicest way to define patterns is with symmetry. So the quantum mechanics that models the elementary particles is defined using the symmetries that are observed in the experiments. While this method is clearly the easiest way of obtaining a model that matches experiment, it should also be clear that the seeking of symmetries is a human attribute and not necessarily an indication of how Nature is most easily understood. Here we explore an alternative way of describing elementary particles.

The Standard Model of elementary particles consists of representatives of SU(3) × SU(2) × U(1) symmetry. Why did nature choose this symmetry rather than, for example, SU(4) × E4 × U(1) × U(1)? The Standard Model provides no explanation for the arbitrariness of the choice of symmetry. On the other hand, what is interesting here is that SU(3) × SU(2) × U(1) is a block diagonal symmetry.

It is not enough to choose the symmetry SU(3) × SU(2) × U(1). The particles are taken as irreducible representations of this symmetry so one must also choose these irreps. To define an irrep of SU(3) × SU(2) × U(1) one chooses an irrep of SU(3), one of SU(2) and one of U(1). Each of these has an infinite number of choices; the ones used by the fermions in the Standard Model are:

SU ( 3 ) SU ( 2 ) U ( 1 ) Particles 1 1 2 right-handedelectron 1 1 0 right-handedneutrino 1 2 1 left-handedleptons 3 1 4 / 3 right-handedupquark 3 1 2 / Algebra of Classical and Quantum Binary Measurements

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