Trace scalar products

A Hermitian operator p is a von Neumann density if it is nonnegative and has unit trace. In more concrete terms, if is the finite-dimensional Fock space for a quantum model where electrons are distributed over a finite number of states, then p is a von Neumann density if (i) v,pv) > 0 for all operators v on and (ii) l)g = 1. By the formula v,pv) we mean the trace scalar product of the operators v and pv, that is, v,pv) = traceg(u pu) since (p, l)g = tracegp = 1 we have used this scalar product to express the trace condition. More generally. 

The superscript 2 is added to emphasize that these interactions are two-body. That is, these operators are orthogonal to all scalar and one-body operators with respect to the trace scalar product. 

Definition 15 A -body operator is a Hermitian operator that can be represented as a polynomial of degree 2 A in the annihilation and creation operators, and is of even degree in these operators. In addition, a A -body operator must be orthogonal to all k — l)-body operators, all k — 2)-body operators,. .., and all scalar operators, with respect to the trace scalar product. 

Recently, a unitarily invariant decomposition of Hermitian second-order matrices of arbitrary symmetry under permutation of the indices within the row or column subsets of indices has been reported by Alcoba [77]. This decomposition, which generalizes that of Coleman, also presents three components that are mutually orthogonal with respect to the trace scalar product [77] ... 

In Liouville space, both the density matrix and the 4 operator become vectors. The scalar product of these Liouville space vectors is the trace of their product as operators. Therefore, the NMR signal, as a function of a single time variable, t, is given by (10), in which the parentheses denote a Liouville space scalar product ... 

Proof. We calculate the scalar product by constructing a linear operator P whose trace is equal to the scalar product. Consider the representation (G, HomCVi, V2), cr) dehned in Proposition 5.12. Let / denote the character of this representation. By Proposition 5.14 we know that / = XiX2- Consider the linear operator... 

The criteria (9.23), (9.24), and (9.26) are all rather obvious properties of Euclidean geometry. All of these properties can be traced back to mathematical properties of the scalar product (R Ry), the key structure-maker of a metric space. We therefore wish to determine whether a proposed definition of scalar product satisfies these criteria, and thus guarantees that M is a Euclidean space. 

For products of two matrices the trace operation is reminiscent of a scalar product it is a sum of indexed quantities just as the scaJar product is a sum of TO indexed quantities. The two-index quantities and Rsr may easily be numbered by the single index ... 

Any mean value, which is the trace of a product of an operator matrix A, say) and a density matrix in the usual orbital basis representation becomes a simple scalar product in the basis-product representation ... 

The trace of a scalar is the scalar itself. Since the inner (dot) product of two vectors xn and y is a scalar, we can write... 

Since a second-rank cartesian tensor Tap transforms in the same way as the set of products uaVfj, it can also be expressed in terms of a scalar (which is the trace T,y(y), a vector (the three components of the antisymmetric tensor (1 /2 ) Tap — Tpaj), and a second-rank spherical tensor (the five components of the traceless, symmetric tensor, (I /2)(Ta/= + Tpa) - (1/3)J2Taa). The explicit irreducible spherical tensor components can be obtained from equations (5.114) to (5.118) simply by replacing u vp by T,/ . These results are collected in table 5.2. It often happens that these three spherical tensors with k = 0, 1 and 2 occur in real, physical situations. In any given situation, one or more of them may vanish for example, all the components of T1 are zero if the tensor is symmetric, Yap = Tpa. A well-known example of a second-rank spherical tensor is the electric quadrupole moment. Its components are defined by... 

Since a second-rank cartesian tensor lap transforms in the same way as the set of products UaVp,it can also be expressed in terms of a scalar (which is the trace Yi aa),... 

---