Non-commutative

On the other hand, as detailed further in Appendix C, if the two properties (F and G) do not commute, the second measurement destroys knowledge of the first property s value. After the first measurement, P is an eigenfunction of F after the second measurement, it becomes an eigenfunction of G. If the two non-commuting operators properties are measured in the opposite order, the wavefunction first is an eigenfunction of G, and subsequently becomes an eigenfunction of F. 

The quantum generalization of the APR Hamiltonian results after supplementing this classical Hamiltonian with a non-commuting angular momentum part [Lj, p] = -ihSji which introduces quantum dispersion and thus qualitatively new effects due to additional fluctuations and tunnehng. 

In this regard, we refer the readers to a few examples of Section 4 in which equality (58) holds true for commutative operators AaAp = Ap Aa-b) If the operators AaP) and A/j t) are non-commutative, then estimate... 

The fact that quantum observables are represented by matrices immediately suggests problems of non-commutation. For instance, the observables can be measured at the same time only if they have a complete orthogonal set of eigenvectors in common. This happens only when they commute, i.e. XY = YX, or the commutator [X, Y] = XY — YX = 0. This is a central feature of the matrix formulation of quantum theory discovered by Heisenberg, Born and Jordan while trying to explain the observed spectral transitions of the hydrogen atom in a more fundamental way than the quantization... 

The fact that px is a semidirect product of these two subalgebras is a necessary condition to support such an interpretation. Indeed, since we have [p,p] = p, we see that the role played by the generators of symmetries p is to impress dynamical modification on the observables p giving rise to other observables. As a consequence, the non-commutativity between the observables is a matter of measurement. In the case we are studying in this section we have [p,p] = p resulting in a quantum theory. For the sake of consistency, we expect to derive a classical TFD theory with an algebra similar to pr but in which [p,p] = 0. (This result has been explored in Ref.(L.M. Silva et.al., 1997))... 

Diagonalization Method to Calculate Expectation Values of Operators Non-Commutative to the Hamiltonian. Vibrational Assignment of HOC1. 

This general non-commutative property of matrix multiplications is in contrast with ordinary algebra. 

In the system of quantum dipoles, dipole and momentum variables have to be replaced by the quantum operators, and quantum statistical mechanics has to be applied. Now, the kinetic energy given by Eq. 9 does affect the thermal average of quantity that depends on dipole variables, due to non-commutivity of dipole and momentiun operators. According to the Pl-QMC method, a quantum system of N dipoles can be approximated by P coupled classical subsystems of N dipoles, where P is the Trotter munber and this approximation becomes exact in the limit P oo. Each quantiun dipole vector is replaced by a cychc chain of P classical dipole vectors, or beads , i.e., - fii -I-. .., iii p, = Hi,I. This classical system of N coupled chains... 

In Eq. (5), the product q q is quaternion-valued and non-commutative, but not antisymmetric in the indices p and v. The B<3> held and structure of 0(3) electrodynamics must be found from a special case of Eq. (5) showing that 0(3) electrodynamics is a Yang-Mills theory and also a theory of general relativity [1]. The important conclusion reached is that Yang-Mills theories can be derived from the irreducible representations of the Einstein group. This result is consistent with the fact that all theories of physics must be theories of general relativity in principle. From Eq. (1), it is possible to write four-valued, generally covariant, components such as... 

Thus the multiplication of quaternion units is non-commutative. In eq. (1) q is to be interpreted as a compound symbol that stands for two different objects the real quaternion,... 

T + T2. Figure 12.6 also shows that the addition of turns is non-commutative ... 

The chronological operator T is required because of the general non-commutativity of an operator taken at two different times [13,16, 17]. 

The partial recovery of the quantum phase coherence of nuclear dipoles originates from the non-commutative property of the Zeeman energy with the quantum operator which represents the residual interaction after rotating the spins. This rotation has no effect on the magnetisation dynamics when the residual interaction, hHR, is equal to zero. No... 

In conclusion we note that the construction of considered models is based on the following property. Their Hamiltonians are the sums of the cell Hamiltonians that are local and non-commuting with each other. At the same time the ground-state wave function of the total Hamiltonian is the ground state for each cell Hamiltonian. It is clear that these models are rather special. Nevertheless, the study of them is useful for understanding properties of the real frustrated spin systems and strongly correlated electronic models. 

This is a very important result because the first term describes the vibrational kinetic energy of the nuclei, whilst the second and third terms represent the rotational kinetic energy. The transformation is straightforward provided one takes proper note of the non-commutation of the operator products which arise. 

The right-hand side of this equation appears to be non-Hermitian since T xp(J) does not commute with 3) p)q((of but the summation over p removes all non-commuting terms. 

Figure 4-3. Illustration for the non-commutative character of the symmetry operations.

It may be that the wave functions are eigenfunctions of two non-commuting operators corresponding to physical quantities such as p (momentum) and q (position) respectively. Then, by measuring either A or B in system I, it becomes possible to predict with certainty and without disturbing the second system, either the value of Pk or qr. In the first case p is an element of reality and in the second case q is an element of reality. But ipk and commuting operators cannot have simultaneous reality. It was inferred that quantum theory is incomplete. 

Again IIW, M° and IM2n M2n, although IMX n + IM2n = M°Xn + M2 n. A product of non commuting matrices always corresponds to an order of decreasing indices, here m, from left to right. Matrices Nn 0 and Nn n are defined as unit matrices. Inversion of eqn. (102) yields... 

W is a complete local ring of characteristic zero with maximal ideal pW, and residue class field W/pW = k these properties of W determine it uniquely (k being perfect), and o is the unique automorphism inducing the automorphism of raising to p-th powers in the residue class field (compare J.-P.Serre -Corps locaux, II.5 6), We denote by A the (non-commutative) ring generated over W by F and V with the relations... 

The ring A is a subring of the ring E of non-commutative formal power series in F and V over W v/ith the same relations explained above (cf. [ 23 ], page 21). We are going to use this result of CARTIER and GABRIEL, and methods of MANIN (cf. C233 ) in order to make some calculations. 

When subscripts 1 and 2 are suppressed, we have the simple case of products of x, y, z], and the six permutationally symmetric products reduce to (4.2), with the three permutationally antisymmetric products vanishing identically when x, y, z are numbers but not necessarily when they stand for (non-commuting) operators. 

As a result of this replacement the linear vibronic interaction in Hamiltonian (1) is cancelled, and the effective virtual phonon interaction operator appears. However the use of this method that formally could be applied to any Hamiltoninan is related to some practical difficulties. These difficulties take place in case of the presence of some non- commuting electron operators in the initial Hamiltonian. They are especially serious when the dynamics of the JT system is under discussion. 

Even if AB and BA have the same order, the two product matrices may not be equal. In these circumstances, matrix multiplication is non-commutative, i.e. AB BA. 

One exception is the commutative law. In general, matrix multiplication is Non-commutative ... 

The effect of applying two sequential coordinate transformations on a point, r, can be represented by the product of the two matrices, each one of which represents the respective transformation. We need to take care, however, that the matrices are multiplied in the correct order because, as we saw above, matrix multiplication is often non-commutative. For example, in order to find the matrix representing an anticlockwise rotation by 0, followed by a reflection in the y-axis, we need to find the product CA (and not AC as we might initially assume ). 

---