Wedge product

An AGP wavefunction [48, 49] is generated from wedge products involving a single geminal g(l,2) ... 

Corrections for the 4-RDM and 5-RDM functionals may be obtained by searching for some terms involving the wedge products of lower ROMs, which cancel with the corresponding corrections for the HRDM functionals [20]. Consider the matrices and A describing the errors in Valdemoro s reconstruction functionals for the 2- and 3-RDMs as well as the matrices A and A describing the errors in Valdemoro s reconstruction functionals for the 2- and 3-HRDMs... 

The theory of cumulants allows us to partition an RDM into contributions that scale differently with the number N of particles. Because aU of the particles are connected by interactions, the cumulant RDMs scale linearly with the number N of particles. The unconnected terms in the p-RDM reconstruction formulas scale between N and W according to the number of connected RDMs in the wedge product. For example, the term scales as NP since all p particles are statistically independent of each other. By examining the scaling of terms with N in the contraction of higher reconstruction functionals, we may derive an important set of relations for the connected RDMs. 

In the contraction of any wedge product the position of the upper and lower indices generates two types of terms in Grassmann algebra [26] (i) pure contraction terms where the upper and lower contraction indices appear on the same component of the wedge product, and (ii) transvection terms, where the upper and lower contraction indices appear on different components of the wedge product. To illustrate, we consider the contraction of the wedge product between A and A ... 

As in the previous section, by connected we mean all terms that scale linearly with N. Wedge products of cumulant RDMs can scale linearly if and only if they are connected by the indices of a matrix that scales linearly with N transvec-tion). In the previous section we only considered the indices of the one-particle identity matrix in the contraction (or number) operator. In the CSE we have the two-particle reduced Hamiltonian matrix, which is defined in Eqs. (2) and (3). Even though the one-electron part of scales as N, the division by A — 1 in Eq. (3) causes it to scale linearly with N. Hence, from our definition of connected, which only requires the matrix to scale linearly with N, the transvection... 

The portion of the 2-RDM that may be expressed as wedge products of lower RDMs is said to be unconnected. The unconnected portion of the 2-RDM contains an important portion of the two-particle component from the unitary decomposition D2, and similarly, the trace and one-particle unitary components contain an important portion of the connected 2-RDM A, which corrects the contraction. Both decompositions may be synthesized by examining the unitary decomposition of the connected 2-RDM,... 

The elements of the matrix may be obtained from and t/ j by summing the distinct products arising from all antisymmetric permutations of the upper indices and all antisymmetric permutations of the lower indices. With the wedge product of one-particle matrices, there are only four distinct possibilities ... 

More generally, we can write the elements of the wedge product as... 

If the sum (p + q) is odd, exchanging the p upper indices with q upper indices will produce a minus sign, but this will be cancelled by another minus sign produced by exchanging the lower indices. In many cases it will be easier and clearer to write the wedge products as in the second form, Eq. (A6), without specifying a particular element through indices. 

The condition that is a positive semidefinite operator ensures that belongs to the Kummer cone. Unfortunately, the Grassmann wedge product of two operators is not explained explicitly in Ref. [1] but the section, p.79 may be sufficient. Here are two other references [4, 5] that may be helpful. Be sure that you understand Exercise 4 on p.73 of Ref. [1]. 

The Yang-Mills functional [17] is defined by the integration of the wedge product F A F, where denotes the Hodge dual-star operator... 

If the gauge connection is Abelian, then the term eA A A vanishes by the antisymmetry of the wedge product. This means that = dAv /. This is an example of an Abelian gauge theory, defined according to that vanishing of commutators between gauge potentials. 

On the other hand the so called wedge product of 1-forms a, p, is an example of a 2-form, at any point in phase space it can be viewed as a quadratie form, i.e. a scalar valued function of two vectors which is linear in each argument. It is written a A P and is defined, for vectors , j R by... 

The last term on the right vanishes by equality of mixed partials and the antisymmetry of the wedge product. On the other hand, using (2.29), we obtain, by similar... 

For simplicity, we will assume the mass matrix is diagonal with ith diagonal element m . Now we write a formula for the time rate of change of the wedge product... 

This follows from the antisymmetry of the wedge product (dx A dy = —dy a dx), and the fact that the Hessian matrix of U is symmetric. Next, note that... 

Using similar argumentation based on the properties of the wedge product, we next show... 

The 2-RDM can be expressed as the wedge product of 1-RDMs (unconnected) plus a cumulant (connected) part [28, 29] denoted as... 

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