Two New Groups

As explained in the main text the role of group theory in quantum mechanics is pivotal since symmetry groups are based on linear algebra which actually governs the symmetry operations, and admits of the existence of gradient invariants that determine both momentum and energy in space and time respectively.

Using k,r ,the first group is capable of describing order-disorder transitions, where there is no change in space group. In the case of an ω, t substitution this O(2) group may be used to describe the photon and the phonon and, since it defines singlet states, it also provides the solution to the EPR paradox. Paired k-state wave functions describe the harmonic oscillations of field vectors for the electric and magnetic fields for the photon.

The gradient invariant, applied to the relevant representation, defines the energy as ωħ.

Representation table for the Lie group O(2)

E

r1,-1

r2,-2

r3,-3

r4,–4

r5,-5

Ө

A1

1

1

1

1

1

1

1

A2

1

1

1

1

1

1

-1

E1

2

2cosk1.r1

2cosk1.r2

2cosk1.r3

2cosk1.r4

2cosk1.r5

0

E2

2

2cosk2.r1

2cosk2.r2

2cosk2.r3

2cosk2.r4

2cosk2.r5

0

E3

2

2cosk3.r1

2cosk3.r2

2cosk3.r3

2cosk3.r4

2cosk3.r5

0

E4

2

2cosk4.r1

2cosk4.r2

2cosk4.r3

2cosk4.r4

2cosk4.r5

0

E5

2

2cosk5.r1

2cosk5.r2

2cosk5.r3

2cosk5.r4

2cosk5.r5

0
Representation table for the Landau group

E

(r1-r5)

Ө

Ë

Ë(r1-r5)

ËӨ

A1

1

11111

1

1

11111

1

A2

1

11111

-1

1

11111

-1

E1

2

2sink1,r1-5

0

2

2sink1,r1-5

0

E2

2

2sink2,r1-5

0

2

2sink2,r1-5

0

E3

2

2sink3,r1-5

0

2

2sink3,r1-5

0

E4

2

2sink4,r1-5

0

2

2sink4,r1-5

0

E5

2

2sink5,r1-5

0

2

2sink5,r1-5

0

B1

1

11111

1

-1

-1-1-1-1-1

-1

B2

1

11111

-1

-1

-1-1-1-1-1

1

E6

2

2cosk1,r1-5

0

-2

-2cosk1,r1-5

0

E7

2

2cosk2,r1-5

0

-2

-2cosk2,r1-5

0

E8

2

2cosk3,r1-5

0

-2

-2cosk3,r1-5

0

E9

2

2cosk4,r1-5

0

-2

-2cosk4,r1-5

0

E10

2

2cosk5,r1-5

0

-2

-2cosk5,r1-5

0

The second group is also based on the Lie group O(2).

This group, using k,r , describes a continuous order-disorder transition, with a change in the space group associated with a Landau Special Point on the surface of the Brillouin zone. Change in the space group involves a situation where a lattice vector in the corresponding direction is doubled on ordering in relation to the Special Point.

A simple lattice point group, of order two, describes the situation involving complete order and complete disorder, where the new operator Ë takes on values of +1 for total disorder and -1 for order, implying that the points concerned are equivalent, since they are both disordered (1), and different since they are ordered (-1)

The direct product of this lattice point ordering group with the group O(2) gives the Landau group.

This new group contains the O(2) group four times and may be characterised in terms of the two parent space groups. The top half of the character table with, Ë = +1 , belongs to the disordered space group, where each lattice point is disordered in A1 and A2. The lower half of the character table with, Ë = -1, in B1 and B2, represents the totally ordered space group.

The behaviour of the group in disordering implies that paired representations are involved. The upper group representations provide sine wave modulation of the disordered space group and the second, cosine modulation is based on the completely ordered space group.

This pairing provides an immediate explanation of the e and f diffraction maxima in an incommensurate phase. The related k vectors are common to both systems.

E        Ë

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