quantum super-operators

quantum super-operators

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This is a contributed topic on

quantum super-operators

(or superoperators).

0.1

Time and Microentropy: Irreversibility in Open Systems

A significant part of the scientific and philosophical work of Ilya
Prigogine has been devoted to the dynamical meaning of phenomenal/physical irreversibility
expressed in terms of the second law of thermodynamics and quantum statistical mechanics.
For systems with strong enough instability of motion the

concept

of phase space trajectories is
no longer meaningful and the dynamical description has to be
replaced by the motion of

distribution functions

on the phase
space. The viewpoint is that

quantum theory

produces a more
coherent type of motion than in the classical setting and the
quantum effects induce correlations between neighbouring classical
trajectories in phase space.

0.2

Quantum Super-operators

Prigogine’s idea (1980) is to associate a macroscopic entropy (or Lyapounov

function

)
with a microscopic entropy (quantum) super–

operator

M

. Here the time–parametrized distribution
functions

ρ

t

are regarded as

densities

in phase space such
that the

inner product

⟨

ρ

t

,

M

⁢

ρ

t

⟩

varies
monotonously with

t

as the functions

ρ

t

evolve in
accordance with Liouville’s equation (Prigogine, 1980; Misra et al, 1979).
For well defined systems for which the super-operators

M

exist, a time super-operator

T

(age or internal time) can also be introduced. (For the precise details, the reader
is referred to Misra et al. 1979). Furthermore, the equations of motion with randomness at the microscopic level then emerge as irreversibility on the macroscopic level. However, unlike the usual quantum operators representing observables, the

M

super-operators are non-Hermitian operators, (that is, they are not self-adjoint,

M

≠

M* ). One notes that Einstein would have discarded such notions along with
the standard quantum mechanics formulation it is interesting that no

imaginary

experiments have been
designed so far to test the reality of quantum super-operators.

However, there are certain provisions that have to be made in terms of the

spectrum

of the

Hamiltonian

H

for M to be properly defined: if

H

has a pure point spectrum, then

M

does not exist, and likewise, if

H

has a

continuous

but

bounded

spectrum then

M

cannot exist. Thus, the super-operator

M

cannot exist in the case of only finitely extended systems containing only a finite number of particles. Furthermore, the super-operator

M

cannot preserve the class of

pure states

since it is non-factorizable. The distinction between pure states (represented by vectors in a

Hilbert space

) and mixed states (represented by density operators) is thus lost in the process of measurement. In other words, the distinction between pure and mixed states is lost in a

quantum system

for which the algebra of observables can be extended to include a new dynamical variable representing the non-equilibrium entropy. In this way, one may formulate the second law of thermodynamics in terms of

M

for quantum mechanical systems. Let us mention that the time operator

T

represents internal time and the usual, secondary time in quantum dynamics is regarded as an average over

T

. When

T

reduces to a

trivial

operator the usual
concept of time is recovered

T

⁢

ρ

⁢

(

x

,

v

,

t

)

=

t

⁢

ρ

⁢

(

x

,

v

,

t

)

, and
thus time in the usual sense is conceived as an average of the

individual

times as registered by the observer. Given the latter’s
ability to distinguish between between future and past, a
self-consistent scheme may be summarized in the following diagram
(Prigogine, 1980):

\xymatrix

⁢

@

⁢

M

=

0.1

⁢

p

⁢

c

⁢

@

=

5

⁢

p

⁢

c

⁢

&

⁢

Observer

\ar

⁢

[

r

]

⁢

&

⁢

Dynamics

\ar

⁢

[

d

]

⁢

&

⁢

Broken time symmetry

\ar

⁢

[

u

]

⁢

&

⁢

Dissipative structures

\ar

⁢

[

l

]

(0.1)

for which irreversibility occurs as the intermediary in the following

sequence

:

Dynamics

⟹

Irreversibility

⟹

Dissipative structures

(Note however that certain quantum theorists, as well as Einstein, regarded the irreversibility of time as an “illusion” caused by statistical averaging. Others– operating with

minimal

representations in

quantum logic

for finite quantum systems– go further still by denying that there is any need for real time to appear in the formulation of quantum theory.)

The importance of the above diagram will become fully apparent in the context of

Section

4 , where we discuss living organisms in terms of open systems that by definition are irreversible, and also have highly

complex

(

generic

) dynamics supported by dissipative

structures

which may have come into

existence

through spontaneous symmetry breaking, as explained in further detail by Baianu and Poli, 2008, in this volume, and also briefly in the next subsection. This diagram sketches four major pieces from the puzzle of the emergence/origin of life on earth, without however coming very close to completing this puzzle; thus, Prigogine’s subtle concepts of microscopic time and micro–entropy super–operators may allow us to understand how life originated on earth several billion years ago, and also how organisms function and survive today. They also provide a partial answer to subtle quantum genetics and fundamental evolutionary dynamics questions asked by Schrödinger– one of the great founders of quantum ‘wave mechanics’– in his widely read book “What is Life?” Other key answers to the latter’s question were recently provided by Robert Rosen (2000) in his popular book “Essays on Life Itself.”, unfortunately without any possibility of

continuation

or of reaching soon the ultimate, or

complete

, answer. Schrödinger’s suggestion that living organisms “feed on negative entropy, or negentropy” was at least in part formalized by Prigogine’s super-operators, such as

M

. This theory is in great need of further developments that he could not complete during his lifespan; such developments may also include several of Rosen ’s (2000) suggestions and will apparently require a categorical and Higher Dimensional Algebraic, non–Abelian theory of irreversible thermodynamics, as well as a quantum–mechanical

statistics

of open systems that are capable of autopoiesis, e.g. living organisms.

References

1

Prigogine, I.: 1980,

From Being to Becoming– Time and Complexity in the Physical Sciences

, W. H. Freeman and Co.: San Francisco.

Title

quantum super-operators

Canonical name

QuantumSuperoperators

Date of creation

2013-03-22 18:12:50

Last modified on

2013-03-22 18:12:50

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

16

Author

bci1 (20947)

Entry type

Topic

Classification

msc 81T99

Classification

msc 81T10

Classification

msc 81T05

Synonym

extension of quantum operator algebras

Related topic

QuantumOperatorAlgebrasInQuantumFieldTheories

Defines

super-operators in extended quantum theory

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