The Vaganov-de Boni Gauge Interpretation

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The Vaganov-de Boni Gauge Interpretation (VdB) is an ongoing interpretive dichotomy in theoretical applied physics of distinct systems of strongly-coupled interacting objects in, for example condensed matter systems. The interpretive nature of the VdB is due to a so-called gauge choice that can be made on the VdB systems, and of which gauge choice is actually physically permitted.

History[edit]

In 1971 the physicist Vassily Vaganov had first observed that there existed a quantum anomaly in the traversal of a fermionic object with respect to its helicity. This effect was first evidenced in an experiment conducted earlier by Daniele Salserio de Boni, who was conducting an experiment on transport fluxes of fully saturated states. de Boni had conducted the experiment following the earlier experiments of Nishimura on saturated states in condensed systems. The effect, once observed by Vaganov, immediately evinced a long-standing implied assumption in all physical systems, even though it was not explicitly apparent before then. The effect that Vaganov first exposed (via de Boni and Nishimura), and that subsequent physicists posed was this: does the newly observed helical anomaly manifest itself universally in all systems, or was it simply a gauge choice? And furthermore, was this a purely gauge effect, in the latter case, or was there some natural choice in the helicity of physical systems. Most subsequent research has found that there exists a broken symmetry in the ground states of traversing systems in strongly-coupled environments. Such systems are observed to favour one helicity state over the other, and the debate of current research is which is the natural state for the system to be in and which is the unphysical ghost state.

Background Physics[edit]

Since, the VdB system is a strongly-coupled effect of interacting objects, we must begin with a description as a quantum field theory (QFT). Thus, let us begin with the fundamentals of quantum field theory, glossing over bosonic states and cutting straight to fermionic systems.

Spinors[edit]

Since all physical systems must be Lorentz covariant, we seek objects that transform under a representation of the Lorentz group. The Lorentz group is defined by the equation:

and with infinitesimal elements, is expressed as:

.

That is, infinitesimally, the Lorentz group is expressed by objects that are antisymmetric in its indices.

An antisymmetric real tensor has 6 independent components. The Lie algebra of the Lorentz group is therefore spanned by 6 generators, which physically correspond to the 3 rotations and 3 boosts of the Lorentz group. Usually, one would index the six generators directly: where . However, it is conventional to instead index the six components as a pair of antisymmetric Lorentz indices: where and . The 6 generators are then straightforwardly defined as:

The generators of the Lie algebra satisfy:

Fermionic physical objects are represented mathematically by objects known as spinors, which are 4-component objects that transform under the spinor representation of the Lorentz group. The spinor representation is spanned by 6 generators that we, again, write by convention as an antisymmetric pair of Lorentz indices:

where

The simplest representation of are of matrices where . For example, one choice for is the chiral representation:

where , and are the Pauli matrices and is the identity matrix.

A spinor is a 4-component object that transforms under a finite Lorentz transformation,

as:

where are 6 numbers that parameterize the transformation, the indices again being a pair of antisymmetric Lorentz indices.

Dirac equation[edit]

The spinors, thus defined, must obey the relativistic energy condition . Quantum mechanically, the energy-momentum operator is , and thus all quantum field theories must satisfy the Klein-Gordon equation:

where we have set and is some field in spacetime. Dirac sought solutions of the "square root" of this equation, forming the Dirac equation:

where the field which satisfies the Dirac equation will automatically also solve the Klein-Gordon equation if

.

Since we have already constructed such objects above, any spinor field that satisfies the Dirac equation is necessarily a construction of a quantum theory of relativistic fermionic objects:

where each component of the spinor field satisfies the energy-momentum constraint.

The four components of the spinor correspond with the four degrees of freedom of a fermion, representing particle and anti-particle states for the two spin states. The solutions of the Dirac equation take the form:

where the spinor takes the form:

where the 2-component spinor encodes the spin of the particle, for example for spin up along z-axis and for spin down along the z-axis . The separation of the 4-component spinor object into two 2-component spinors is because the spinor representation is reducible. Under rotations (parameterized by ) and boosts (parameterized by ), the spinor transforms via matrices:

and

.

Since the transformations of the Lorentz group are of block diagonal form, the representation is reducible, clearly, two 2-component spinor representations. The representation sifference is evident under boosts, whereby the two representations boost in opposite directions even though they both rotate in the same direction. That is, if we write

then the 2-component spinors and transform under the block diagonal matrices above. By convention, the spinor is said to transform in the representation of the Lorentz group, and the spinor is said to transform in the representation; the spinors are related by a parity transformation.

The two plane-wave solutions above, if interpreted as particle states would correspond to positive and negative energy states because of the factor or . But interpreted instead as a Fourier decomposition into plane wave modes, the spinor field can be written:

,

where we decompose the spinors defined above into basis spinors where representing the two fundamental spin states. Furthermore, we label the independent plane-wave solutions above as and for the positive and negative plane-wave energy states. Thus, all told, we have a specific spinor labelled as and with its spinor index explicit: .

The decomposition of into plane-wave states has Fourier coefficients and respectively. We interpret these as ladder operators. The conjugate operator is given by:

The ladder operators obey anti-commutation relations since fermions must obey the Pauli Exclusion Principle:

and every other anti-commutator vanishes; e.g., .

The operator creates a particle of 3-momentum and spin (where is spin-up, say, and is spin-down). The operator creates an anti-particle of 3-momentum and spin (again, is spin-up, say, and is spin-down):

Helical Anomaly[edit]

In a strongly interacting system, for example in systems of densely packed interacting particles, on average the motion of particles is random unless an external interaction is applied in a privileged direction. When this happens, classically, one would expect that there would be a transport pressure applied to the system, indiscriminately of the state of the individual particles. However, quantum mechanically, the transport coefficients are apparently affected by the helicity state of the transported particles. The action of an interacting system can be modelled as:

where for simplicity the model assumes that only one particle is being transported () and there is no backreaction to the system of stable immobile particles (). The interaction between fermions is mediated, here, by Yukawa couplings with some generic scalar particle ().

The system is placed in a -dimensional setting, but we will only be interested in - a two-dimensional (plus time) system. It is in this system that the helical anomaly is present. In higher dimensions the effect is negated by the extra degrees of freedom of the system.

The interaction term is parameterized by a coupling strength, and also a neighbouring function, , which encodes the fact that the particle is interacting most strongly with its nearest neighbours. Whilst the neighbouring function is physically a complex term, we can simplify the model by taking the system to be a square lattice (other choices can also be made, but don't change the final result).

The important thing to note is that the model does not assume that the particles are identical: they have different self-interaction terms . We can rewrite the interaction-free terms by relabelling and then:

The difficulty lies in the fact that the interaction term cannot be written as

since the transport particle does not interact with all particles in the system. The trick to apply here is that we write the system as if this were the case and then include a Lagrange multiplier () to encode the fact that the system is not as free as it prima facie appears. In the Lagrange multiplier term, we enforce the privileged direction of the transport explicitly:

This system now has a conformal symmetry and is invariant under transformations of , but only in the unbroken directions; that is, those that are not privileged under the direction . We can make this more explicit by separating the broken states from the symmetric ones:

where we have implicitly relabelled the unbroken states as to not clutter the equations, and the broken states have been written with the projection onto the index is via the generator that picks out the privileged direction:

and where is an group element. Note that the broken state has free propagation terms similarly projected:

.

We now dualize this strongly-coupled system by using the Buscher-Witten procedure on the terms and . This will transform the system to a weakly-coupled system from which we can perform a perturbative analysis. More specifically, it means we can then discard higher-order terms in , which we cannot do here. The Buscher-Witten procedure involves promoting the Lagrange multiplier terms into full dynamical variables. This is physically possible because the broken Lagrangian terms admit a gauge symmetry:

.

It was univerally thought that this gauge symmetry would render the system indiscriminant to the state of the particle projections. So, while the system has a broken symmetry along the transport direction (), which means that particles are more prone to move along than perpendicular directions, the transport would not be affected by the state of itself. However, because the system is strongly coupled the full dynamics of the system are not explicitly known: it was only thought that this symmetry protected the system from the state of itself. By dualizing the system, it was made clear that, in fact, this is not the case. This is the helical anomaly.

The Buscher-Witten procedure involves gauging the terms as normal, but then eliminating the gauge by integrating out the gauge paramters and promoting the Lagrange multiplier to a dynamical variable. When this is done, the action takes the form:

In this action we see the extra terms induced by the gauging of - we can observe the new dynamical fields in . The important observation is that Yukawa coupling has been modified to:

Since the transport state has a vacuum expectation value aligned along the transport direction , the first order corrections to will effectively keep the parameter small, and hence pertubatively valid. Important also is the fact that the Buscher-Witten procedure has promoted the Lagrange variable to the actual transport velocity of the system . This is because the procedure involves integrating out the gauge terms and the presence of the Dirac delta function means that the Lagrange multiplier picked out the transport direction. Since the Yukawa term now includes the transport direction itself in its perturbative expansion, it means that how the state transforms under Lorentz boosts will affect the gauge coupling.

Since the parity state of now affects its transport (since spinors transform differently under boosts depending on if it is left- or right-parity symmetric), the quantum system should see a difference in transport depending on its helicity. The only open question is which of the states is favoured in this regard since it is not clear how the full perturbative interaction ultimately affects its transport.

An Analogy[edit]

A person (red) wishes to move along the direction of a crowded table in restaurant. The people in the way (black) act like a strongly coupled system of interacting particles impeding the path of the motive particle. The fundamental system is an interacting one of different particles; in the analogy, this represents people of different sizes that impede the path of the motive person.
The motion of a person will naturally align themselves either butt-to-butt or butt-to-face to the static people. The question remains which is the natural alignment of people (particles).

To help understand what is going on, let us use an analogy. Suppose there is a busy restaurant and there are many people seated along a large table. Since the restaurant is busy, it is not necessarily easy for individual people to move about. In this scenario, we therefore are necessarily talking about a strongly interacting system. Suppose now that one person decides to get out and moves along the direction parallel to the table, sliding along the other seated people (physically, an external potential is applied to the system picking out a direction along which the particle moves).

You would think that no matter what the moving person does, it would be difficult for them to get out: they can't do anything about their own state that will affect how they move past the others. However, if you think about it, the person moving out naturally does align themselves with the people they are moving past in order to get out as quickly as possible. They do this by aligning themselves, naturally, by facing along with or against the people they are moving against. In other words, they align themselves typically either butt-to-face or butt-to-butt, but never any other orientation. They naturally align themselves in one direction or another because they inherently feel moving aligned in that particular direction is more natural. Consider for yourselves whether you have ever done this and ever had to explicitly think about what you chose to do. The Vaganov-de Boni gauge interpretation is that a system of particles will naturally align themselves in one direction or another. The only open question is whether for systems of particles (people) it is more natural (and physically correct) for people to move aligned butt-to-butt or butt-to-face, as it were.

The literature is divided on the natural alignment that is to be taken because there may be other strong-coupling considerations that are important in the choice to be taken. These were pointed out by de Boni first, soon after Vaganov had observed the presence of the effect and others had informally settled on the naturally correct alignment, as far as they saw it.

Here follows some selected opinions that might weigh in on the correct choice to be made by the people (particles). For convenience, the helicity-aligned opinion is referred to as "butt-to-face" and the helicity-anti-aligned opinion is referred to as "butt-to-butt". Note also that the quotes are edited to fit the analogy presented above for clarity purposes.

I believe that... actually the [people] should be [aligned] butt-to-butt. The risk of strong coupling effects between the travelling [person's] state and the stationary [person] is disastrous. Consider that the [travelling person's] spin state, whilst initially aligned in the z-direction may over time acquire a VEV in the x-direction. This means that the spin of the travelling person is in complete danger of a disastrous interaction with the [stationary people]. This would be unacceptable to the system and thus the [people] must align butt-to-butt to avoid this at all costs.

Dr D. S. de Boni, Physica Status Solidi B 90, K127 (1978)

In my opinion, the only physically correct orientation of [people] is for the moving [person] to be aligned butt-to-face. In this orientation, the motion of the [person] is near-frictionless because, being identically orientated, both states are aligned and there is no possibility that there will be any interference of one [person] on the others, other than bulk thermal effects. This is regardless of the types of strongly-interacting effects proposed by de Boni because, ultimately, I believe it would be more [uncomfortable (the phrase actually used was "disruptive")] for [people] to be aligned butt-to-butt. Remember that we are talking about systems of fermions, which don't like to [face one another (actual phrase used: "be in the same state as one another"].

Dr. Atrak Nokoer, Phys. Rev. B 31, 2147 (1985)