connection with Stark Hamiltonians, Hehenberger et al. [23], Brändas and Froelich
[24] and for Floquet Hamiltonians, Howland [25].
Even if we have greatly oversimplified the details of the spectral decomposition,
the dilatation analytic theory covers the whole analytic family of Hamiltonians and
provides important testing grounds for extended investigations of dynamical
quantum chemical systems, which goes beyond traditional approaches of quantum
mechanics, Nicolaides and Brändas [19]. It will be shown below that most of the
quantum mechanical machinery will remain intact in accounting for the “move
into” the complex energy plane, nevertheless allowing for a much more general set
of solutions, usually denoted “unstable states in the continuum”, at the same time
giving rise to a more realistic contractive time evolution. However there is an
important difference, i.e. the traditional use of matrix transformations via unitary
transformations must be generalized to educe complex symmetric similitudes. As a
consequence one might encounter situations where the Hamiltonian matrix cannot
be diagonalised. This is usually considered to be a nightmare in contemporary
numerical analysis, but is here a blessing in disguise, allowing important conceptual
notions and fundamental abstractions.
6 Statistical Mechanics far from Equilibrium—Off-Diagonal
Long-Range Order
Our objective is here to incorporate these so-called “unstable states” in a more
general non-equilibrium quantum statistical framework. Obviously our formulation
must go beyond the Maxwell-Boltzmann distribution and the traditional FermiDirac or Bose-Einstein statistics. According to Nernst’s theorem, sometimes called
the third law of thermodynamics, a perfect crystal has zero entropy in the limit
when the absolute temperature T ! 0. Slightly generalized we could use a similar
argument for any system with a nondegenerate ground state denoted by a properly
antisymmetrised N-particle (fermionic) wavefunction W. From this follows a
sequence of N-representable reduced fermionic density matrices, see e.g. Löwdin
[26]
C
ðqÞ x 1 ; x 2 ; . . .x q jx
0
1 ; x
0
2 ; . . .x
0
q
¼
N
q
Z
Wðx 1 ; x 2 ; . . .x q ; x qþ1 ; . . .x N ÞW
à ðx
0
1 ; x
0
2 ; . . .x
0
q ; x qþ1 ; . . .x N Þdx qþ1 ; . . .dx N
ð6:1Þ
A Zero Energy Universe Scenario: From Unstable Chemical …
261
[24] and for Floquet Hamiltonians, Howland [25].
Even if we have greatly oversimplified the details of the spectral decomposition,
the dilatation analytic theory covers the whole analytic family of Hamiltonians and
provides important testing grounds for extended investigations of dynamical
quantum chemical systems, which goes beyond traditional approaches of quantum
mechanics, Nicolaides and Brändas [19]. It will be shown below that most of the
quantum mechanical machinery will remain intact in accounting for the “move
into” the complex energy plane, nevertheless allowing for a much more general set
of solutions, usually denoted “unstable states in the continuum”, at the same time
giving rise to a more realistic contractive time evolution. However there is an
important difference, i.e. the traditional use of matrix transformations via unitary
transformations must be generalized to educe complex symmetric similitudes. As a
consequence one might encounter situations where the Hamiltonian matrix cannot
be diagonalised. This is usually considered to be a nightmare in contemporary
numerical analysis, but is here a blessing in disguise, allowing important conceptual
notions and fundamental abstractions.
6 Statistical Mechanics far from Equilibrium—Off-Diagonal
Long-Range Order
Our objective is here to incorporate these so-called “unstable states” in a more
general non-equilibrium quantum statistical framework. Obviously our formulation
must go beyond the Maxwell-Boltzmann distribution and the traditional FermiDirac or Bose-Einstein statistics. According to Nernst’s theorem, sometimes called
the third law of thermodynamics, a perfect crystal has zero entropy in the limit
when the absolute temperature T ! 0. Slightly generalized we could use a similar
argument for any system with a nondegenerate ground state denoted by a properly
antisymmetrised N-particle (fermionic) wavefunction W. From this follows a
sequence of N-representable reduced fermionic density matrices, see e.g. Löwdin
[26]
C
ðqÞ x 1 ; x 2 ; . . .x q jx
0
1 ; x
0
2 ; . . .x
0
q
¼
N
q
Z
Wðx 1 ; x 2 ; . . .x q ; x qþ1 ; . . .x N ÞW
à ðx
0
1 ; x
0
2 ; . . .x
0
q ; x qþ1 ; . . .x N Þdx qþ1 ; . . .dx N
ð6:1Þ
A Zero Energy Universe Scenario: From Unstable Chemical …
261