The calculations for the NNLO and higher corrections to the snuark mass interaction are difficult. I’ve already got some sort of sign error. Before I go on, I’m going to write a computer program to do the higher level calculations.
Of course it is possible, and even natural, to write a Java program to do snuark QFT calculations. One can implement the Pauli algebra projection operators for spin in the +/- x, y, and z directions, and multiply away. One computes the phases by writing the product of projection operators, for example, as a complex constant k times the final and initial projection operators: , or in Pauli matrix form:
In the same manner, arbitrarily long products of projection operators can be reduced to a complex multiple of the initial and final projection operators. If there is a forbidden transition somewhere in that product, then the complex multiple will be zero. For the above product, .
There are several convenient ways of making these calculations. The one I will discuss here, and the one that I will use in the computer program, will use the “fictitious vacuum” of Julian Schwinger and makes the calculation with creation and annihilation operators similar to those used in the standard model. This makes for a very fast and easy to write computer program. It also gives a good comparison of the strengths and weaknesses of the geometric QFT that I use versus the standard QFT.
Complex Numbers in QM
Both QM and QFT are methods of producing a model of the physical world that computes first a complex number, and then takes the squared magnitude of the complex number to get a real number. This real number gives the probability of the interaction. It is very easy to get a PhD in physics without realizing this. For the lay person, an excellent reference is Feynman’s magnificent and inexpensive ($4.84 used on Amazon) book QED: The Strange Theory of Light and Matte. While this book uses no mathematics, those who know mathematics will have little problems putting the equations back in. Feynman is that good.
We will use the familiar bra-ket notation. In the case of QM, the complex number is computed as a complex product of two vectors, for example, the probability for transitions from the state A to the state B is given by first computing the complex number , and then taking the squared magnitude of this complex number. Quantum mechanics arises from upgrading the position and momentum data for a classical particle to operators. Quantum field theory does the same thing, more or less, with classical fields. In doing this, both theories reduce a calculation to a complex number.
Those of us who play with geometric algebra or geometric calculus try to find geometric replacements for the imaginary numbers of physics. As an example, the Pauli algebra is written as a vector of three operators, which in our convenient notation we call x, y, and z. If we were to write these out as Pauli spin matrices, our “representation” would use imaginary numbers, but these are intrinsic only to the representation. One could ditch the imaginary numbers by using real matrices though this would increase the size of the matrices to 4×4 instead of 2×2. In the Pauli matrices, the product of all three matrices is the imaginary unit matrix:
but from a geometric algebra point of view the above product is xyz, a purely geometric object.
The lack of complex numbers doesn’t stop geometric algebra from making computations that are analogous to those made the usual way. The discussion is too long for a blog post, (and is not necessary for the non perturbative calcultions we are working on here), but if you want to know the details, see Chapter 2 “Geometry” of my incomplete book on density matrices.
In short, the complex numbers we will be dealing with here are geometric quantitites that arise naturally from the Clifford algebra, not the representation of the Clifford algebra. Here we will treat the imaginary unit as just that, but eventually, when we distinguish between the charged and neutral leptons, we will have to dig deeper.
The Vacuum State
In QFT, as in QM, one must reduce a calculation down to a complex number. This is done in the context of a Hilbert space, a vector space with a complex inner product.
If one wrote QFT without a vacuum, one would have to deal with complicated inner products such as: where a, b, and c are a varying number of initial particle states and x, y, and z are a varying number of final particle states.
Instead of doing QFT this very obvious way, one notices that the Hilbert space includes vectors corresponding to various numbers of particles. Since the Hilbert space is a mathematical object (and not a physical experiment), we invent operators that magically add and subtract particles without any other effects. These are the creation and annihilation operators, and we use them to replace the initial and final particle states with operators acting on a vacuum state. By using them, we can rewrite the inner product as follows:
I believe that the original usage of the vacuum state was by physicists working to apply quantum mechanics to crystals. In such a situation, the vacuum state is the quantum state of the crystal with no phonons. The creation and annihilation operators change the number of photons. Several important recent QFT textbooks (Zee’s Quantum Field Theory in a Nutshell comes to mind) explain the theory from this point of view.
The Measurement Algebra
Snuark theory is based on Julian Schwinger’s The Algebra of Microscopic Measurement. In the measurement algebra, we associate an operator M(a) with the quantum state a. His notation is somewhat obscure, but if you follow it carefully you will find that it is a generalization of the usual density matrices. For the qubits, these are the projection operators.
Schwinger notes that in general, a measurement may change the quantum state. He writes measurements of this sort M(a,a’) where a’ is the initial state and a is the final state. In our work we have avoided this notation as it leads to problems with the assigning of complex phase to these objects. So long as a and a’ are not completely incompatible measurements, one can uniquely assign complex phases by assigning M(a,a’) = M(a)M(a’), that is, by multiplying the density matrices. However, when a and a’ are incompatible, such as spin up and spin down, the product M(a)M(a’) becomes zero and there is no natural assignment of complex phase.
Mathematical aside: The lack of a natural assignment of complex phase is related to a not very widely known fact about spinors. The various directions in 3-space can be defined by the points on the surface of the unit sphere. For each of these directions, one can define a quantum state. For instance, the point (0,0,1) and (0,0,-1) define spin up and spin down, respectively. If one represents the quantum states as normalized spinors, then each spinor is defined only up to a complex phase.
It is impossible to define this complex phase in a continuous manner. The problem with assigning complex phases only applies to those who attempt to choose phases for ALL the spinors. It is possible to make a map that is continuous at all but a single point (and we will do so below). For the snuark case, there are only six spinors of interest; we can easily define their phases arbitrarily.
The astute reader is invited to write a short proof of the above statement into the comment section. Preferably at an algebraic level that this author can understand.
If there were a continuous assignment of the complex phases of Schwinger’s general measurement symbols M(a,a’), then it would be possible to define a continuous choice of phase in the spinors. To see this, write the M(a,a’) as 2×2 matrices using the Pauli spin matrix representation. Let a’ be spin up. Then the left column of M(a,a’) is precisely the 2×1 spinor |a>. Thus M(a,a’) defines the complex phases of the spinors.
Defining M(a,a’) = M(a) M(a’) works for all but the case when a and a’ are completely incompatible, which happens only at one point. Therefore, this gives a method of defining the phase of the spinors that is continuous at all but one point. Let (a,b,c) be a unit vector. The vector for spin in the (a,b,c) direction is s = ax+by+cz. The density matrix for this state is (1+s)/2 but 1+s is easier to compute with. Converting this to 2×2 matrices using the Pauli algebra gives 1+s =
The left column is a spinor for spin in the (a,b,c) direction. It is continuous and nonzero except at (a,b,c) = (0,0,-1). Normalizing this spinor, we have a continuous normalized spinor solution for all spin in the (a,b,c) direction except spin down, where there is an essential singularity:
The right column is a spinor for spin in the (a,b,c) direction nonzero except at (0,0,1).
A Fictitious Vacuum
Schwinger defines a “fictitious vacuum” in his second paper on the measurement algebra, The Geometry of Quantum States. He does this by adding an extra state to the measurement algebra, one that does not correspond to any real state. He calls this state 0, and defines the creation operators as M(a,0) since these operators take the vacuum state and turn it into state a, and the annihilation operators as M(0,a) for similar reasons.
Since spin up and spin down are a complete set of states in standard quantum mechanics, one typically writes problems in these states only. If one needs a state with some other orientation, one can write it as an appropriate superposition of the spin up and spin down states. This form of calculation makes it particularly natural to use Pauli spinors (which come from a representation of SU(2) with the z spin operator diagonalized). What was an elegant geometric language (the Pauli algebra) becomes ugly tensors.
We are using +x, +y, and +z, that is, (1,0,0), (0,1,0), and (0,0,1), to represent the left handed chiral states. We will choose the fictitious vacuum as the midpoint of the spherical triangle defined by these vectors, . While this state is not “fictitious” in that it is a valid quantum state, we can treat it as such. This choice of fictious vacuum is only useful to us to the extent that we restrict our attention to states that contain only a single quantum state. Fortunately, this is the nature of the calculations we need to make. What makes it particularly convenient is that it allows us to write our creation and annihilation operators inside the Pauli algebra.
Products of (primitive) projection operators such as the density matrix quantum states, can be reduced to complex multiples of the initial and final projection operator (providing that these two projection operators do not annihilate each other). If both the initial and final projection operators are spin +1/2 in the (1,1,1) direction, the result will be a complex multiple of the vacuum state. Since the algebra of the complex multiples of an idempotent is equivalent to the algebra of complex numbers, we will call such a thing a complex number and write it in bra ket form, for example as
If only the initial projection operator is (1,1,1), Schwinger calls such a product a creation operator. If only the final operator is (1,1,1), he calls it an annihilation operator. We will follow his terminology.
Geometry versus Creation and Annihilation
The strength of geometric QFT is that the vertex amplitudes are computed from geometry alone. While we’ve been using a vertex amplitude of “iq” we will eventually show that this is a number that we can derive from an assumption of equal transition probabilities, and in doing this, we will see that q is what we expect. In the geometric theory, complex numbers appear when we reduce complicated geometric product to the final operator times the initial operator, all multiplied by a complex constant.
This is not what Schwinger does in his papers that use the general measurement symbols. Schwinger is more interested in the traditional study subjects of QFT, things having to do with dependency on position, rather than the qubit applications discussed in this blog. With general measurement algebra symbols, complex phases are suppressed and the equivalent calculations produce no amplitude. For example, let M(+y,+x) and M(+z,+y) be the general measurement symbols for the conversion of a +x snuark to +y, and a +y snuark to the +z state. Then the general measurement symbols give M(+z,+y) M(+y,+x) = M(+z,+x).
Particle Masses: A few posts ago we looked at long lived snuark bound states and found a set of three coupled quadratic equations. In solving these equations, we found that there were three particle states implied and associated these with the three particle generations.
The above calculations looked only at the long term amplitudes. We completely ignored the details of where these amplitudes came from. In particular, we did not use any geometric calculations in working these out. We looked only at long term amplitudes. We assumed that the amplitudes for a time period of length T and a time period of length 2T were the same, and from this, and the usual convolution of Feynman diagrams, found the quadratic equations. These were three coupled quadratic equations over three complex values.
In making these calculations we removed the geometry from the problem and were dealing with amplitudes that no longer have geometric phases that show up when they are multiplied. We would have obtained the same calculations if our bare mass interaction had no geometric phase at all. This would be the case if we used creation and annihilation operator type calculations.
We will get masses from them by summing up the squares of the real values (or traces) of all the contributing Feynman diagrams. This amounts to computing the contribution to the particle energy from the various activities that are going on. But to do this, we have to calculate the energy of the underlying Feynman diagrams, with corrections. This will bring in geometric factors. If we neglected this correction, we would be assuming that the mass interaction has no geometric phases or amplitudes related to it; the particle masses would depend only on the long term amplitudes and not at all on the details of the bare interaction, with corrections.
Alternatively, we could put the geometric factors back into the Feynman diagram convolution. This would slightly complicate the quadratic equations as they would include complex amplitudes. On solving these equations, the results will be the same, provided that one notes that the trace of the off diagonal interactions (i.e. 1/4) is half the trace of the diagonal ones (i.e. 1/2).
The same lack of complex amplitudes in products occurs when one brings in a fictitious vacuum. One computes: M(0,+z) M(+z,+y) M(+y,+x) M(+x,0) = M(0,0). This would be different from the geometric method because it gives no complex amplitude. To get the geometric result in vacuum form, we have to assign complex constants to the M(+z,+y) and M(+y,+x) vertices. Writing things out in a Feynman form, we have:
In other words, to convert geometric calculations into creation and annihilation operator form, we have to compute a complex amplitude for each mass interaction vertex.
Examining the algebra results at the bottom of the above diagram, we see an odd feature. In the geometric calculation, the complex amplitude only shows up when you merge two mass interactions into one. Separately, the two mass interactions have no intrinsic geometric amplitude. By contrast, in the creation and annihilation calculation, each mass interaction will have to have its own complex amplitude intrinsic to itself; the overall amplitude will be the product of the two amplitudes.
More generally, in a product of N projection operators, the geometric method gives the overall effective amplitude as a product of N-1 complex numbers, while the creation and annihilation operator method will give the effective amplitude as a product of N complex numbers. The difference in the number of products is accounted for by the fact that the reduced geometric product (the product of the final and initial operators), when written in vacuum form, will have yet another complex number associated with it. To get the complex amplitude, we take the N vacuum complex numbers and divide by this last complex number. Counting powers, this gives an overall power of N-1, just as the geometric calculation gives.
There are six elements in the snuark algebra, spin in the +x, +y, +z, -x, -y, and -z directions. To fully specify the vacuum equivalent mass interactions, we have to give amplitudes for 6×6 = 36 interactions. We will write these as for example, for the +y to +x amplitude.
In using to mean the complex number for the +y to +x interaction rather than the +x to +y interaction, we are following matrix multiplication where the initial state is on the right and the final state is on the left. This is an enduring source of confusion for the author, who in the best of circumstances tends to drop minus signs, so please beware.
The 6 identical interactions like +x to +x are trivial as the projection operators are idempotent, and so the geometric and vacuum amplitudes are the same and k is 1. The 6 interactions that change sign, such as +x to -x or -y to +y, are forbidden, k is 0. This leaves 24 interactions.
Our choice of the vacuum state, spin in the (1,1,1) direction, is convenient in that it is symmetric with respect to +x, +y, and +z. Thus the are symmetric with respect to cyclic permutations. This reduces the number of k we have to calculate by a factor of 3 leaving 8 to compute. Furthermore, reversing the sequence causes a complex conjugate, that is, . We are now down to just 4 computations: .
Our previous calculations of geometric phases allows us to derive a formula for the constant. The +x to +y transition takes the same value of k as the +y to +z, and these are the complex conjugate of the +z to +x transition. (As mentioned above, this follows from our choice of a vacuum state symmetric with respect to x, y, and z.) We have:
Writing , for the magnitude we have and for the phase we have where n is an integer. Therefore for n = 0, 1, or 2. This calculation is how I came to guess the formula for the neutrino masses which has attracted some attention.
In the next post we’ll evaluate the various .