My Current Research

My final important step in the development of incommensurate theory involved an appreciation of the fact that it must be compatible with translational symmetry, specifically in relation to atom displacement in the order-disorder processes existing in the plagioclase feldspars and elsewhere. In a paper on the incommensurate behavior in the plagioclase feldspars, in the Can. Min. (2008) , 46 1389-1400, I used the pair of conventional Abelian exponential translation matrices exp ± ik.r, in a similarity transformation, to obtain the pair of real orthogonal Lie SO(2 ) group matrices which together form the continuous Lie group O(2). Here r and k= 2π/λ are vectors, and k,r is simply an inner product which yields a phase angle in the range 0 to 2π.

These Abelian matrices have determinant +1. They form the basis of the representation tables for two new translation groups described in my paper of 2008. Matrices (1) and (2) are inverse and hence describe translations in direction ± r respectively They combine to form the non-Abelian Lie group O(2). Note, for future reference, that all the possible matrices in the SO(2) Lie groups commute. Also, that Eugene P. Wigner, in Group Theory, 1959 Academic Press, Chapter 14 developed the rotational dihedral groups in exactly the same manner. In the present case the paired Abelian components of each of these matrices describe the probability amplitudes of two independent functions associated with cosine and sine terms, and thus provide a pair of k wave functions defining displacements in terms of the general field of quantum mechanics. However, in this version of quantum mechanics we use the cosine function initially to define position on displacement, and the sine function to describe the associated gradient of position.

At this point we note that our new mathematical definition of quantum behaviour, as related to translational symmetry, makes no reference to either mass or velocity and may be adapted for use in a wide range of stationary problems including order-disorder and incommensurate behaviour.

In the latter case we must now understand immediately the significance of the independent e and f systems of reflections, already described, in the plagioclase feldspars. They represent the two interacting order-disorder functions, cos (k,r) and sin (k,r) which our formative mathematical theory links to the Abelian SO(2) Lie groups.

In restructuring our stationary version of quantum theory it is now appropriate to consider the gradient invariant which has already been discussed. It is central to defining the energy of interaction of the independent paired k wave functions as dictated by translational symmetry. A group theoretical definition is necessary and is provided in (3):

The gradient invariant is therefore equal to ±k for the alternate subgroups with energy equal to kħ where ħ is the reduced version of Planck’s constant. Note that the gradient invariant determinant (GID) demands the existence of two functions, and that the conventional use of a single wave function in quantum mechanics is unacceptable since it cannot provide translational invariance.

At this point we must establish the nature of the k-vector distribution where a pair of interacting and independent wave functions are in gradient relationship with one another. The combination of SO(2) matrices (1) and (2) and the gradient invariant (3) , which, incidentally provides a proof of Noether’s first theorem, provide the basis of a generalised quantum theory. In this version the paired k wave functions are independent, and particles are represented simply as particles. The k-wave system also exists independently of the single crystal, and defines permissible active modes for any chosen dynamic effect. In practice it defines the subgroup modes of an infinite crystal. If the linear dimension of the single crystal is L-1, in terms of the number of lattice points, permissible modes have wavelengths given by the set L-1/n where n is an integer, implying that the reciprocal wavelength is n/L-1. This means that the individual dynamic modes must form a completely regular reciprocal lattice in one, and in three dimensions. The total number of k vectors in the associated Hilbert vector space is necessarily limited by the volume of the Brillouin zone. This ensures that the number of k vectors is equal to the number of lattice points, and that the final symmetry group representation tables are square.

The first problem in applying our revised quantum theory to order-disorder, is the identification of the relevant functions described by our SO(2) matrices. It is important to note that this choice is defined by the nature of the space group itself, as we determined from the dual incommensurate k-wave systems e and f already discussed. This implies that we must use the Landau special points in the Brillouin zone to define, as products, separate order and disorder modulations. The disorder special point is usually represented by the origin spectrum 000. The two functions chosen are gradient related as required

Since the new n-state dynamic reciprocal lattice implies that the k-space energy is necessarily linear in nkħ it follows, in terms of free energy, that the resulting order parameter, defined in terms of the sum over nk terms, must also be linear, as a function of temperature, for a one-dimensional system. However, since the density of k states must increase as T³, within a three dimensional Brillouin zone, the disorder parameter must also behave as T³. Since it is usual instead to define the order parameter as β, this implies that the exponent of the order parameter must be ⅓ as observed in a vast number of recorded order-disorder transitions. This condition also applies to the transition in superfluids This simple quantum mechanical solution to the unsolved problem of order-disorder transitions in three dimensions also provides the necessary correction to Landau’s order-disorder theory, which provided a mean field solution only, yielding an incorrect β exponent of one-half. We must still utilize Landau’s introduction in relation to the concept of special points for ordering however.

The origin of incommensurate behavior in order-disorder situations, as in the plagioclase feldspars, may now be explained quite simply. In the middle of a solid solution system, where the end members must order independently, and factors such as the incorrect Al/Si ratio in tetrahedral sites must exist, it is inevitable that a single phase solution in ordering must involve residual disorder. Hence the normal ground state of complete order is inaccessible and the single phase in ordering must adopt a low entropy solution in which order and disorder are combined. The residual disorder in this case must be described in terms of the highly degenerate incommensurate single k-state solution associated with maximum usage of the lowest accessible k-state ordering mode. This constitutes a Fermi condensation and provides an important new feature of Fermi statistics which actually negates the exclusion principle.

The results of a calorimetric study of the ordering energy of incommensurate phases in the plagioclases were published in Geochim. et Cosmochim. (1985) Acta 49, 947-966. They showed that the incommensurate state was indeed extremely well ordered. In the hydrothermal laboratory my colleague Michael Carpenter was also able to demonstrate that this minimal and highly degenerate state occurred reversibly, as a first order transition, at a temperature of 800^o Centigrade, Carpenter M.A. (1986) Phys. Chem. Minerals 13, 119-139.

Ignoring the essential symmetry criteria in quantum mechanics associated with translation has other important consequences. A very interesting example is the existence of the Einstein Podolsky Rosen paradox, which relates to the apparent interaction of particles at a distance from one another. We start the explanation of apparent long range interaction by developing the Lie group O(2) of Table (1) from its two subgroups, as already defined in terms of the matrices (1) ands (2). Since they are inverse, they define displacements in opposite directions, and must also have inverse properties. They come together to form singlet states in the Lie group O(2), as proved by using a similarity group transformation with the inversion operator, which we describe as Ө, in (4).

These two SO(2) anti-symmetrical matrices are necessarily degenerate and must fall into the same class, in the same representation, in the non-Abelian Lie group O(2). We now propose a situation in which a reaction occurs with the appearance of two photons in a single O(2) singlet class. The total momentum is zero since their momentum cancels. As they move apart they must also continue to have an anti-symmetrical relationship with each other. Hence it is quite clear that the initial k-state symmetry relationships must be retained indefinitely. This conclusion also extends to the detailed cosine and sine components of the combined photon k-wave system. This implies that the local harmonic oscillation of both photons must oscillate in exact anti phase in both space and time regardless of the separation. By the way, the group behaviour in time in equations (1), (2) (3) and (4) is simply described by substituting ωt for k,r where ω =2 π/τ and the time interval τ is equal to λ/c in the case of the photon. We assume that cyclic conditions must also occur in time as well as space and that Table (1) may be modified accordingly, in order to deal with mobility. The gradient invariant determinant (3), in this case, yields the energy ħω.

I think that Einstein might have been happy with this simple solution to the apparent paradox, particularly since this version of quantum mechanics is fully deterministic.

At this point we link to the pair of new groups and their explanation  

At this point it is perhaps reasonable to ask why, in the last ninety or so years, no one, apparently, has considered the essential role of translational symmetry groups in the context of particles in motion, and particles that are dynamically displaced. I believe that a proper appreciation of this situation relates to a major separation between the relevant mathematics and physics. Group theory is an abstract application of linear algebra which need not have any relationship to practical problems. However it is invaluable where one is able to combine serious mathematical insight with a very real knowledge of the nature of a chosen physical problem, with a particular interest in the origin of symmetry itself.

The important function of symmetry groups in quantum mechanics is dependent on the fact that they are based on linear algebra. This linear property is essential, in the context of gradient invariance on translation, as we have demonstrated in equation (3). Thus it follows that the mathematics associated with symmetry must always be particularly simple. As Dirac is reputed to have said, if it is not simple it is probably wrong.

At this point I have some comments on Schrödinger’s quantum equation. Since he uses the exponential term exp ik.r, the gradient invariant must be ik which relates to angular momentum proportional to k. However, as already noted, the gradient invariant (3) for translation is simply k and hence this single exponential term is unfaithful. Finally, terms in angular momentum do not necessarily commute, leading to a situation involving the uncertainty principle, which is unacceptable.

Finally we discuss the matter of publication. Since the material which I have set out above provides a new slant on quantum mechanics, and represents a paradigm shift towards a more formal group theoretical approach, conventional journals and referees have reacted as one might expect. Mathematicians rightly state that the groups are trivial but they do not necessarily understand the physical implications. Condensed matter physicists do not see the point of the exercise, and in one case (prl) both referees were completely unaware of the fact that a prior solution to the three-dimensional order-disorder problem did not exist. The solution is to present my results in a personal web site. It is already apparent that the new Landau quantum group, which models change in the space group in the presence of an order-disorder transition, has the necessary group theoretical potential to describe the superconducting and superfluid transitions, which is the point that my current research has now reached.