2 Time Evolution, Partitioning Techniqueand Associated Dynamics
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6
E.J. Brăandas
(t), respectively, we can separate out positive and negative times (with respect to
an arbitrary chosen time t = 0) as related with the relevant contours C± according to
G± (t) = ±(−i) θ (±t)e−iHt ; G(z) = (z − H)−1
(1.2)
where the retarded-advanced propagators G± (t) and the resolvents G(z± ); z± =
Rz ± iI z are connected through (R z, I z are the real and imaginary parts of
z = E + iε )
+∞
1
±
−izt
G(z)e dz; G(z) = G± (t)eizt dt
G (t) =
2π
C±
{t > 0; I z > 0}
−∞
{t < 0; I z < 0}
(1.3)
The Fourier-Laplace transform Eq. 1.2 exists under quite general conditions by e.g.
closing the contours C± in the lower and upper complex planes respectively. More
details regarding the specific choice of contours in actual cases can be found in
Refs. [13–15] and references therein. The formal retarded-advanced formulation
corresponding to Eq. 1.1 including the memory terms follows from
i
∂
− H G± (t) = δ (t); ψ ± (t) = ±iG± (t)ψ ± (0)
∂t
i
∂
− H ψ ± (t) = ±iδ (t)ψ ± (0)
∂t
(1.4)
and
(z − H)G(z) = I; ψ ± (z) = ±iG(z)ψ ± (0)
(z − H)ψ ±(z) = ±iψ ± (0)
(1.5)
ψ ± (t),
It is usual to normalize the time dependent wavefunction
which means that
ψ (z) obtained from partitioning technique as a result is not. Since we are primarily
interested in the case E ∈ σAC we will take the limits I (z) → ±0, obtaining the
dispersion relations
G(E + iε ) = lim (E + iε − H)−1 = P(E − H)−1 ± (−i)πδ (E − H)
ε→ ± 0
(1.6)
where P is the principal value of the integral.
The goal is now to evaluate full time dependence from available knowledge of the
wave function ϕ at time t = 0, for simplicity we assume that the limits t → ±0 are
the same, although this is not a necessary condition in general. Making the choice
O = |φ φ |φ −1 φ |; φ = ϕ (0), one obtains (the subspace, defined by the projector
O can easily be extended to additional dimensions)
ψ (0) = ψ + (0) = ψ − (0); Oψ (0) = ϕ (0); Pψ (0) = κ (0); O + P = I
(1.7)
1 Time Asymmetry and the Evolution of Physical Laws
7
Using familiar operator relations of the time dependent partitioning technique, see
again e.g. Ref. [15] for a recent review, we obtain
O(z − H)−1 = O(z − O H (z)O)−1 (I + HT (z))
H (z) = H + HT(z) H; T (z) = P(z − PHP)−1 P
(1.8)
As we have pointed out at several instances the present equations are essentially
analogous to the development of suitable master equations in statistical mechanics
[4–7], where the “wavefunction” here plays the role of suitable probability distributions. Note for instance the similarity between the reduced resolvent, based on
H (z), and the collision operator of the Prigogine subdynamics. The eigenvalues of
the latter define the spectral contributions corresponding to the projector that defines
the map of an arbitrary initial distribution onto a kinetic space obeying semigroup
evolution laws, for more details we refer to Ref. [6] and the following section.
Rewriting the inhomogeneous version of the Schrăodinger equation, where the
boldface wave vectors below signify added dimensions, one obtains (the poles of
the first line of Eq. 1.8 correspond to eigenvalues below)
(z − H)Ψ (z) = O(z − H (z))Oφ
(1.9)
From Eq. 1.9 the formulas of the time-dependent partitioning technique follows
straightforwardly
ϕ ± (t) = ±
i
2π
O(z − H)−1 ψ (0) e−izt dz = ±
C±
i
2π
ϕ (z)e−izt dz
(1.10)
C±
where
ϕ (z) = O(z − H)−1 ψ (0) = O(z − OH (z)O)−1 (ϕ (0) + HT(z)κ (0)
(1.11)
The equations of motion, restricted to subspace O, is directly obtained from the
convolution theorem of the Fourier-Laplace transform, i.e.
i
∂
− OHO ϕ ± (t) = ±iδ (t)ϕ (0)
∂t
⎫
⎧
⎬
⎨ t
(GP (t − τ )PH ϕ (τ ))± d τ ± iG±
(t)κ (0)
+OHP
P
⎭
⎩
0
(1.12)
with
−iPHPt
G±
=
P (t) = ±(−i)θ (t)e
1
2
T (z)eizt dz
C
(1.13)
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E.J. Brăandas
and
t
(GP (t )PH ( ))± d τ = ±
0
i
2π
T (z)H ϕ (z)e−izt dz
(1.14)
C±
The first term on the right in Eq. 1.12 yields the description of the amplitude ϕ ±
evolving according to the Hamiltonian OHO. Furthermore the second term depends
on all times between 0 and t, while the last term evolves the unknown part κ at t = 0
completing the memory at initial time. Sofar no approximations have been set and
no loss of information acquiesced.
Introducing the auxiliary operator G±
L (t), through
+∞
±
±
iz t
G±
dt
L (t)e
OH (z )O =
−∞
and the non-local operator G± (t; 0) via
t
(GL (t − τ )ϕ (τ ))± d τ = ±
±
G (t; 0)ϕ (0) =
0
i
2π
OH (z± )Oϕ (z)e−izt dz (1.15)
C±
one gets the more compact expression
i
∂ ±
ϕ (t) − G± (t; 0) ϕ (0) = ±i δ (t)ϕ (0) + OHPG±
P (t)κ (0)
∂t
(1.16)
Note that analogous evolution formulas hold within the subspace P, i.e. with
O and P interchanged. Although any localized wave packet under free evolution
disperses, it is however traditionally recognized that the complete formulation
of an elementary scattering set-up describes a time symmetric process provided
the generator of the evolution commutes with the time reversal operator and the
time-dependent equation imparts time symmetric boundary conditions. Compare
for instance analogous discussions in connection with the electromagnetic field,
obeying Maxwell’s equations, via retarded-, advanced- or symmetric potentials.
Although time symmetric equations may exhibit un-symmetric solutions via specific
initial conditions the fundamental point here concerns the “master evolution equation” itself. Hence, as already pointed out, we re-emphasize that no approximations
have been admitted and consequently time evolution proceeds without loss of
information.
At this junction it is common to discuss various short time expansions and/or
long time situations, i.e. to consider partitions of relevant time scales. We will
principally mention two interdependent scales, i.e. a global relaxation time τrel and
a local collision time τc . For instance, during τc the amplitude ϕ is not supposed to
1 Time Asymmetry and the Evolution of Physical Laws
9
alter very much. Hence one approximates the convolution in Eq. 1.16 for positive
times, i.e.
G+ (τc ; 0) ≈ OHO − iτc {OHPHO + 1/2!(−iτc)OH(PH)2 O · ·}
(1.17)
The relaxation time τrel obtains from time independent partitioning technique.
Accordingly starting from Eqs. 1.5 and 1.6 one attains the limits
(E − H)ψ ± (E) = ±iψ (0)
+∞
±
−1
ψ (E) = ±i lim (E + iε − H) ψ (0) = lim
ε →±0
ε →±0
−∞
ψ ± (t)ei(E+iε )t dt (1.18)
Incidentally we recover the stationary wave, ψc (E), via the causal propagator Gc (t)
ψc (t) = Gc (t)ψ (0) = ψ + (t) + ψ −(t); Gc (t) = e−iHt ;
(1.19)
and formally
+∞
ψc (E) = lim
ε →+0
−∞
+∞
ψ + (t)ei(E+iε )t dt + lim
ε →−0
−∞
+∞
ψ − (t)ei(E+iε )t dt =
ψc (t)eiEt dt.
−∞
Since the relaxation or life time, τrel , is directly related to a “hidden” complex
resonance eigenvalue of Eq. 1.9, or pole of Eq. 1.8, we need to derive and investigate
the dispersion relation for the reduced resolvent T (z), i.e.
lim T (E + iε ) = lim (E + iε − PHP)−1
ε →±0
ε →±0
= P(E − PHP)−1 ± (−i)πδ (E − PHP)
(1.20)
For instance, in the none-degenerate case one finds by getting the real and imaginary
parts of f (z)
f (z) = φ |H (z)|φ
(1.21)
f ± (E) = fR (E) ± (−i) f1 (E)
(1.22)
and
with
fR (E) = E + φ |HP(E − PHP)−1 H|φ
fI (E) = π φ |H δ (E − PHP)H|φ ≥ 0
(1.23)
In the limit ε → ±0, assuming full information for simplicity at the initial time,
t = 0, i.e. κ (0) = 0; φ = ϕ (0) = ψ (0), Eq. 1.9 yields
(E − H)Ψ ± (E) = fI (E) (0)
(1.24)
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E.J. Brăandas
keeping in mind that z = E − iε from Eq. 1.23
E = fR (E); ε (E) = f1 (E)
(1.25)
Note that Eq. 1.25 only gives the resonance approximately. The exact complex
resonance eigenvalue (if it exists, see more on this below) has to be found by
analytical continuation, Eq. 1.9, by e.g. successive iterations until convergence for
each individual resonance eigenvalue. Hence in the first iteration, one gets the
lifetime τrel given by
f1 (E) = ε (E) = {2τrel (E)}−1
(1.26)
To find an “uncertainty-like” relation between the two time scales we combine the
expansion Eq. 1.17 with
OH (z)O = OHO +
1
1
OHPHO + OH(PH)2 O · ·
z
z
and
OH (E ± i0)O = OHO + OHPP (E − PHP)−1 PHO
±(−i)π OHPδ (E − PHP)PHO
obtaining
τc σ 2 (E) = ε (E) =
1
2 τrel
(1.27)
with σ2 being the variance at E0 = ϕ (0)|H|ϕ (0) , i.e.
σ 2 = ϕ (0)|HPH|ϕ (0) = ϕ (0)|(H 2 − ϕ (0)|H|ϕ (0) 2 )|ϕ (0)
(1.28)
The relations above contain many approximations, e.g. break-up of convolutions,
truncation of various expansions etc., not to mention the assumed existence of a
rigorous analytical continuation into higher order Riemann sheets of the complex
energy plane. Since the original Schrăodinger equation, or Liouville equation, as
pointed out, is time reversible and rests on a unitary time evolution, it is obvious that
the present lifetime analysis is contradictory. This is a well-known fact amongst the
practitioners in the field; nevertheless, a lot of physical and chemical interpretations
have been made and found to be meaningful portrayals of fundamental experimental
situations. How come that this still appears to work satisfactorily? In the next section
we will examine the reasons why, as well as develop the necessary mathematical
machinery to rigorously extend resolvent- and propagator domains and examine its
evolutionary consequences.
The object of our description is twofold: first to show that analytic continuation
into the complex plane can be rigorously carried out and secondly to examine
1 Time Asymmetry and the Evolution of Physical Laws
11
the end result for the associated time evolution. Not only will we find that time
symmetry is by necessity broken, but also that novel complex structures appear
with fundamental consequences for the validity of the second law as well as giving
further guidelines on a proper self-referential approach to the theory of gravity.
1.3 Non-self-Adjoint Problems and Dissipative Dynamics
Staying within our original focus in the introduction, we recapitulate the current
dilemma, i.e. how to connect, if possible, the exact microscopic time reversible
dynamics, see above, with a time irreversible macroscopic entropic formulation
without making use of any approximations whatsoever. In the present setting one
must ascertain precisely what it takes to go from a stationary- to a quasi-stationary
scenario. To begin with, we return to the practical problem of extracting meaningful
life times out of the exact dynamics presented above and in particular to dwell on
the consequences if any.
The understanding, interpretation and practical tools to approach the problem of
resonances states in quantum chemistry and molecular physics are basically very
well studied. Generally one has either (i) concentrated on the properties of the
stationary time-independent scattering solution (ii) attempted to extract the Gamow
wave by analytic continuation and/or (iii) considered the time-dependent problem
via a suitably prepared reference function or wave-packet. In each case the analysis
prompts different explanations, numerical techniques and understanding, see e.g.
Ref. [15] for a review and more details.
In order to appreciate the significance of this situation, we will portray one of the
most significant and successful approaches to quasi-stationary unstable quantum
states by re-connecting with the previously mentioned theorem due to Balslev
and Combes [3]. The authors derived general spectral theorems of many-body
Schrăodinger operators, employing rigorous mathematical properties of so-called
dilatation analytic interactions (with the absence of singularly continuous spectra).
The possibility to “move” or rotate the absolutely continuous spectrum, σAC ,
appealed almost instantly and was right away exploited in a variety of quantum
theoretical applications in both quantum chemistry and nuclear physics [16].
The principle idea stems from a suitable change, or scaling, of all the coordinates
in the second order partial differential equation (Schrăodinger equation), which if
allowing a complex scale factor, permits outgoing growing exponential solutions,
so-called Gamow waves, to be treated via stable numerical methods without being
forced to leave Hilbert space. Although this trick admits standard usage of alleged
L2 techniques, there is a price, i.e. the emergence of non-self-adjoint operators
which brings about a lot of important consequences to be summarized further below.
The strategy is best illustrated by considering a typical matrix element of a
general quantum mechanical operator W (r) over the basis functions, ϕ (r) and
φ (r), where we write r = r1 , r2 , . . . rN ; assuming 3N fermionic degrees of freedom.
12
E.J. Brăandas
Employing the scaling r = 3N r; η = eiϑ (or η = |η |eiϑ ), where the phase ϑ ≤ ϑ0
for some ϑ0 that in general depends on the operator, one finds straightforwardly
ϕ ∗ (r)W (r)φ (r)dr =
∗
ϕ ∗ (r )W (r )φ (r )dr
(1.29)
or in terms Dirac bra-kets (with ϕ ∗ (η ∗ ) = ϕ (η ))
ϕ |W |ϕ = ϕ (η ∗ )|W (η )|φ (η )
(1.30)
We assume that the operator W (r) as well as ϕ (r) and φ (r) are properly defined
for the scaling process to be justified. For simplicity we also take the interval of the
radial components of r to be (0, ∞). Note that Eq. 1.29 contains the requirement that
the matrix element should be analytic in the parameter η , demanding the complex
conjugate of η in the “bra” side of Eq. 1.30. This is the reason why many complex
scaling treatments in quantum chemistry are implemented using complex symmetric
forms.
In order to appreciate the fine points in this analysis, we therefore return to the
domain issues, i.e. how to define the operator and the basis functions so that the
scaling operation above becomes meaningful. Following Balslev and Combes [3],
we introduce the N-body (molecular) Hamiltonian as H = T + V , where T is
the kinetic energy operator and V is the (dilatation analytic) interaction potential
(expressed as sum of two-body potentials Vij bounded relative Tij = Δij , where the
indices i and j refers to particles i and j respectively). As a first crucial point we
realize that the complex scaling transformation is unbounded, which necessitates a
restriction of the domain of H; note that H is normally bounded from below. Hence
we need to specify the domain D(H) of H as
D(H) = {Φ ∈ h, H Φ ∈ h}
(1.31)
where h denotes the well-known Hilbert space. The essential property of a dilatation
analytic operator is that each individual pair potential of the interaction V is bounded
relative the corresponding part of the kinetic energy. Hence the unboundedness is
due to the latter i.e. D(H) = D(T ). With these preliminaries one can prove that the
scaling operator U(ϑ ) = exp(iAθ ) is unitary for real ϑ and generated by
A=
1 k=N
∑ [pk xk + xk pk ]
2 k=1
(1.32)
where xk and pk are coordinate and momentum vectors of the particle k. As a result
we get
U(eϑ )Φ(r) = exp(iAθ )Φ(r) = e
3Nϑ
2
Φ(eϑ r)
(1.33)
and with ϑ → iϑ ; η = eiϑ , or more generally η = |η |eiϑ we write
H(η ) = U(η )H(1)U −1(η ) = η −2 T (1) + V (η )
(1.34)
1 Time Asymmetry and the Evolution of Physical Laws
13
At this stage it is crucial to emphasize that the formal expression Eq. 1.34 must
be obtained in two steps due to the unboundedness of T and the complex scaling
transformation. First we introduce Ω = {η , | arg (η )| ≤ ϑ0 } in agreement what
has been said above, then decompose Ω in its upper and lower parts, partitioned
by the real axis R, where Ω = Ω+ ∪ Ω− ∪ R and R = R+ ∪ R− ∪ {0}. To avoid
problems we will exclude the point {0}. The first step consists of the real scaling, i.e.
η ∈ R+ , which corresponds to a unitary transformation, followed by an analytical
continuation to η ∈ Ω+ , corresponding to a similarity, non-unitary operation. Since
this is an important point we will consider the scaling operator U(η ); η ∈ Ω+ more
exactly by bringing in the dense subset N (Ω)
N (Ω) = {Φ, Φ ∈ h; H(η )Φ ∈ h; U(η ) ∈ h; η ∈ Ω}
(1.35)
as the well-known Nelson’s class of dilatation analytic vectors [17] more specifically
defined as follows. A vector φ ∈ D(A) is an analytic vector of A if the series
expansion of eAϑ φ has a positive radius of absolute convergence, i.e.
∞
∑
n=0
An φ
n!
ϑn < ∞
for some ϑ > 0. For our purpose, to be explained below, we introduce the Hilbert
(or Banach) space norm
ϑ0
sup
η ∈Ω
U(η )φ < ∞;
U(η )φ
2
L2
dϑ = φ
2
N (ϑ0 )
(1.36)
−ϑ0
To include the kinetic energy operator in our discussion it is natural to request that
the first and second partial derivatives should also satisfy Eq. 1.36 hence introducing
(i)
the spaces Nϑ0 with i = 0, 1, 2 analogously.
With these preliminaries we can now make the precise definition of the selfadjoint analytic family H(η ) as
H(η ) = U(η )H(η )U −1 (η ); D(UHU −1 ) = Nϑ20
H(η ) = η −2 T (1) + V(η ); D(UHU −1 ) → D(T )
(1.37)
Recapitulating, the first step consists of restricting the Hilbert space to a smaller
(2)
domain Nϑ0 for which the scaling U is defined for all complex η values with its
arguments smaller in absolute value than ϑ0 . The second step, after the parameter
ϑ has been made complex (ϑ → i ϑ ), consists of completing the Nelson class of
dilation analytic vectors to the domain of H or in this case T . Here this means
convergence with respect to the standard L2 norm (for both the functions and its
first and second partial derivatives).
14
E.J. Brăandas
To appreciate the reason for our painstaking carefulness at this particular stage
we come back to our frequent references to the fundamental dilemma expressed
above and in the introduction. First the theorem of Balslev and Combes provides us
with a rigorous path into the second Riemann sheet of the complex energy plane.
The factor, η −2 , appearing in front of the kinetic energy operator T , see Eqs. 1.34
and 1.37, has a simple and natural effect. It means that the absolutely continuous
spectrum of H(η ) is rotated in the complex plane with a phase angle equal to −2ϑ .
In this process complex resonance eigenvalues become “exposed” in agreement with
the aforesaid generalized mathematical spectral theorem [3]. However, there is a
small price to be paid, viz, in the process of the analytic continuation the two steps
mentioned above entails a small inevitable loss of information represented by the
restrictions necessary for the definition of the whole analytic family of the operators
H(η ). As we will see this will have consequences both for the entropic as well as the
temporal evolution. These results, all the same, guarantee that the approximations
made in the previous section could be meaningful despite our words of warning.
There are in effect two principal consequences that we will examine. Firstly
the spectral generalization [3] in terms of appearing complex poles of the actual
resolvent, with the complex part interpreted essentially as the reciprocal life-time of
the state and secondly the dynamical outcome regarding the time evolution, i.e. the
conversion of an isometry to a contractive semigroup [18].
To appreciate the first generalization, i.e. modifying the projection operator
formulations of Sect. 1.2, the following construal is supplied
T (η ; z) = P(η )(z − P(η )U(η ) H(1)U −1 (η )P (η ))−1 P(η )
O(η ) = |φ (η ) φ (η ∗ ) |; P(η ) = I − O(η )
φ (η ∗ ) |φ (η ) = φ (1) |φ (1) = 1
(1.38)
Apart from giving the impression of being a rather formal extension, there are two
important points to consider. First the projectors are oblique, i.e. idempotent
O2 (η ) = O(η )
but not self-adjoint
O† (η ) = O(η ∗ ) = O(η )
Furthermore the present bi-orthogonal construction authorize non-probabilistic
formulations allowing e.g. the possibility of zero norms, viz. from Eq. 1.9 one may
encounter, starting with a none-degenerate eigenvalue, that
Ψ (η ∗ ; z∗ ) |Ψ (η ; z) = 1 + Δ(η ; z) = 1 − f (z) = 0
(1.39)
The emerging singularity is associated with a degeneracy of so-called Jordan-block
type, an abysmal situation in matrix theory; see e.g. Ref. [18] and references therein.
In our case, as we will see, this will actually be a “blessing in disguise.” In passing
1 Time Asymmetry and the Evolution of Physical Laws
15
we note that the observed loss of information carries an unexpected increase of
entropy. The occurrence of Jordan blocks, see more below about associated spectral
degeneracies and their interpretations, implies that full information as to the given
state becomes uncertain at the “bifurcation point”, with an associated entropic
increase as a result.
The second consequence regards the dynamics. As already pointed out the stepwise approach is of basic relevance for the use of dilatation analytic Hamiltonians
as generators of contractive semigroups. The problem of comparing classical and
quantum dynamics and the appropriate choice of so-called Lyapunov converters
were examined in some detail in Ref. [8]. Briefly we will review the implication
as follows. Consider an isometric semigroup, cf. the causal propagator in (1.19),
G(t);t ≥ 0, defined on some Hilbert space h. If there exists a contractive semigroup
S(t);t ≥ 0 and a densely defined closed invertible linear operator Λ, with the domain
D (Λ) and range R (Λ) both dense in h, such that
S(t) = ΛG(t)Λ−1 ; t ≥ 0
(1.40)
on a dense linear subset of h, then Λ is called a Lyapunov converter. A necessary
condition for the existence of Λ for a given G(t) = e−iHt is that the generator H has a
non-void absolutely continuous spectral part, i.e. σAC = ∅. In view of what has been
said above it is natural to ask whether H(η ) generates a contractive semi-group, i.e.
that (note that the + -sign in S+ is not a “dagger”)
S+ (t, η ) = U(η )G(t)U −1 (η ) = e−iH(η )t ; t ≥ 0
(1.41)
This is indeed true for many types of potentials, but unfortunately not for the case
of the attractive Coulomb interaction. Although the Balslev-Combes theorem for
dilation analytic Hamiltonians guarantee that the modified spectrum lies on the
real axis (bound states) and in a subset of the closed lower complex halfplane,
a further requirement (using the Hille-Yosida theorem) is that the numerical range
must also be contained in the lower part of the complex energy plane. In addition
the 1/r potential is problematic both at the origin and at infinity; see e.g. Ref. [19]
for a detailed treatment of resonance trajectories and spectral concentration for a
short-range perturbation resting on a Coulomb background. Here a resonance in
the continuous spectrum carries typical ground-state properties [20] and allows for
complex curve crossings (Jordan blocks) [21]. Since the long-range Coulomb part
in a many-body system will be screened by the other particles the anomalies of the
Coulomb problem should not be crucial with respect to the isometric-contractive
semi-group conversion in realistic physical systems. For additional discussions on
this point, involving a slightly more general definition in terms of quasi-isometries
see Ref. [8].
Summarizing; the loss of information, i.e. restricting the full unitary time
evolution to an isometry (weak convergence) before the conversion via a suitable
Lyapunov converter to a contractive semi-group (strong convergence) is an objective
process in contrast to the subjective preparation of any initial state involving various
16
E.J. Brăandas
levels of course graining. It is also important to realize that the completion of a
dense subset of Hilbert space with respect to the appropriate norm gives different
limits depending on whether it is carried out before or after the conversion, hence
we will speak of an informity rule, i.e. a certain natural loss of information, which
is compatible with broken temporal symmetry.
Finally, on account of the impairment of information loss, it is all the same
important to mention that, in the classical as well as in the quantum case, it is
not enough to conclude that the mere existence of a Lyapunov converter explains
or derives time irreversibility and, in the Liouville formulation, guarantees the
approach to equilibrium [8]. In the next section we will concentrate on the
degenerate state before moving on to the relativistic situation looking for the only
remaining explanation of irreversibility in terms of a formulation involving a spatiotemporal dependent background.
1.4 The Jordan Block and the Coherent Dissipative Ensemble
As already mentioned the informity rule prompts several consequences one being
the emergence of so-called Jordan blocks or exceptional points. Although belonging
to standard practise in linear algebra formulations we will proffer some extra time
to this concept. In addition to demonstrate its simple nature we will also establish
a simple complex symmetric form not previously obtained, see e.g. Refs. [11, 14,
21, 22]. Let us start with the 2 × 2 case, where it is easy to demonstrate that the
Jordan canonical form J and the complex symmetric form Q are unitarily connected
through the transformation B, i.e.
Q = B−1 JB = B† JB
(1.42)
where
Q=
1
2
1 −i
−i −1
; J=
0 1
0 0
1
; B= √
2
1
i
1 −i
(1.43)
We note that the squares of Q and J are zero, yet the rank is one. Although these
Jordan forms do not appear in conventional quantum mechanical energy variation
calculations they are not uncommon in extended formulations. For various examples
of the latter instigated in quantum physical situations, we refer to [11] and also to
applications of a new reformulation of the celebrated Găodel(s) theorem(s) in terms
of exceptional points, Ref. [12]. Note that the alternative formulation in terms of an
antisymmetric construction is not anti-hermitean as wrongly indicated in Ref. [12].
As demonstrated, complex symmetric forms are naturally exploited in quantum
chemistry and molecular physics and therefore we need to extend Eqs. 1.42 and
1.43 to general n × n matrices. The mathematical theorem that a triangular matrix is
similar to a complex symmetric form goes back to Gantmacher [23], but the explicit