Download Effective action in quantum generalization of statistical

Effective action in the quantum
generalization of Equilibrium
Statistical Thermodynamics
A.D.Sukhanov,
O.N.Golubjeva
1
References
• O. Golubjeva, A. Sukhanov. Some geometrical
analogues between the describing of the states space
in non-classical Physics and the events space in
classical Physics. VI Intern. Conference BGL-08.
Debrecen, Hungary, 2008 (in print)
• A. Sukhanov. Some consequences of quantum
generalization of statistical thermodynamics . XIII
Intern. Conference “Problems of quantum field theory”.
Dubna, Russia, 2008 (in print)
• A. Sukhanov. A quantum generalization of
equilibrium statistical thermodynamics: effective
macroparameters. TMP, 154 (1), 2008
2
• A. Sukhanov. Schroedinger uncertainties relation
for a quantum oscillator in a thermostat. TMP,
148, 2 . 2006
• A. Sukhanov. Quantum oscillator in a thermostat
as a model thermodynamics of open quantum
systems. Physics of particles and nuclei. 36, 7a,
2005
• A. Sukhanov. Schroedinger uncertainties relation
and physical features of correlated-coherent states.
TMP, 132 (3), 2002
• A. Sukhanov. On the global interrelation between
quantum dynamics and thermodynamics. XI Int.
Conf. “Problems of quantum field theory”.
Dubna, Russia, 1999
3
Contents
• Introduction
• Operator of effective action
• Correlate-coherent states and effective
action vector
• Effective action as a macroparameter
• Effective action for QO in quantum
thermostat
• Effective action and thermodynamic Lows
• On the Third Low of Thermodynamics
• Conclusion
• References
4
Introduction
• In the last years there appeared some
additional experimental and theoretical
reasons showing that the minimal value of
entropy in thermodynamic equilibrium is
not equal to zero. Particularly, in the
quantum generalization of equilibrium
statistical thermodynamic (QEST) recently
supposed by us the effective entropy
minimal value is
5
• But this fact is in contrary to the standard
result of usual quantum equilibrium
statistical mechanics (QESM)) according
to it entropy must be equal to zero at the
limit T0.
• Let me shortly remind line of reasoning
that give such result in QEST. In our
theory a quantum thermostat at an
effective temperature T* is introduced. It
characterizes quantum and thermal
influence of environments simultaneously
6
• In its frame the effective entropy for
quantum oscillator in thermal equilibrium
is expressed by a new macroparameter –
the effective action J as
• For introduction of the quantity J in our
previous paper we used some heuristic
consideration from classical mechanics
(particularly – notion of adiabatic
invariants) allowing us in fact to postulate
it in the form
7
• where
is the energy of QO in thermal equilibrium
(according to Planck). We see that at T 0 the
effective action for QO runs up to its minimal
value
and
8
• We emphasize that the approach used in
QEST is a macroscopic one. Besides it
was only approved on a model QO. As a
result there exists some dissatisfaction.
We think it would be better to have more
firm evidences in support of this step, i.e.
of introduction of quantity J that has a
deep physical meaning. Besides we
would like to extend QEST ideas to
another microobjects in thermal
equilibrium.
9
• That is why, we are trying to look at this
problem on the other hand. Now we are
starting to move from a microscopic theory
to confirm that we are on the correct way.
• We proceed from the assumption that so
far apart approaches as micro- and macrotheories,
can give us agreed results.
• After this introduction let me pass to the
matter.
10
Operator of effective action
• If we know a wave function (q) that is
usually for microscopic approach we
have a possibility to directly count the
quantity J determining the corresponding
operator on the Hilbert space of states
but not to postulate it! As some heuristic
reason for this goal can serve an analysis
of mutual dependent fluctuations of
coordinate and momentum on the base of
Schroedinger uncertainties relation (SUR).
11
As is well known their dispersions are given by
formulas
•
=
;
=
where
•
and
are fluctuations operators of coordinate and
momentum. Next we use the CBS-unequality in
respect to states submanifold
• The subject of our special interest is the term in
the right side of the expression.
12
• The quantity
has a sense of an transition amplitude from the
state
to the state
. At the same time it
is the Schroedinger correlator and equals the
mean value of an operator
in an
arbitrary state. Considering that the quantity R
has non-zero value in any non-classical theory
(like QM, ST,etc ) we can claim that the given
operator is meaningful. Taking also into account
its dimension we can thereafter call it “effective
action” operator
13
• Of course, we remember operators
and
are non-commutative. Besides their
product is non- Hermitian one.
• Taking the operator in the form
we can select here the two Hermitian terms
•
and
where
is the unique operator
14
• It is easily to see that the operator
reminds the expression of standard
fluctuations correlator of coordinate and
momentum in the classical probabilities
theory. It reflects a contribution of
stochastic environment influence into the
transition amplitude Rpq because it
includes the both fluctuations operators.
That is why we will call it operator of
“ external effect”. We remark that it was
earlier discussed by Schwinger.
15
• At the same time the operator
does not
have any analogies out of quantum dynamics. It
reflects the universal feature of microobject "to
feel" stochastic influence of “cold” vacuum and
to react under it. It is proportional to unique
operator. It means that any state
transforms
to the same state under its using. Therefore it
can be called "own action” operator . Its mean
value does not depend on the state and
always has the constant value for any
microobject
16
• We are of opinion, all this information is a
sufficient reason to claim that we have
deal with an universal concept. We
suggest to give it the special name - the
own action. Its very value is already
eloquent because it is connected with
such well known quantity as the
fundamental Planck constant
.
17
• However as some remark we would like to
tell you some words of terminology
because usually one calls
“elementary quantum of action”. But in J0
there is the meaningful coefficient ½.
And we know that there is not a half of
quantum. Hence, in our opinion, J0 and
might have some different physical
sense and different names.
18
The effective action as a
macroparameter.
• It is obviously that the mean value of operator
coincides with the transition amplitude R pq and it is
a complex quantity
with a phase
.
Its modulus
we consider as macroparameter and call simply
"effective action" .
19
• In quasi-classical limit (at
) Rpq
goes to the real quantity. In this limit
operators
and
can be changed
by c-numbers so  becomes correlator
from the classical probabilities theory.
• The significance of quantity J is in the fact
that it is of paramount important in the
right side of SUR "coordinate-momentum"
20
Correlate-coherent states and
effective action vector
• In a number of important cases this
expression takes a form of strict equality
(the saturated SUR)
• The subject of our interest in this talk are
states in which the saturated SUR are
realized
21
• If at the same time  0 we have deal
with correlate-coherent states (CCS).
Among such states one can select a family
of thermal CCS. They describe
microobjects in thermal equilibrium.
• Besides there exist states called the
simplest coherent states (CS) for that the
term
is absent ( for example, the basic
state of oscillator at T=0 or the initial state
of wave packet).
22
• We can pass from each this CS to CCS
using (u,v)- Bogoliubov transformations
with parameters depending either on
temperature or on time. The given
transformations generate Lie-group
SU (1,1) which is local isomorphic to
Lorenz-group in two-dimensional events
space.
23
• It means we can consider a set (J,) as twodimensional time-like vector in some pseudoEuclidian space and the quantity
as a
length of the vector or the invariant of
corresponding group
It is easily to see that effective action "vector" (J,)
is an analog of two-dimensional vector energymomentum (
, p ) in relativistic mechanics
(For more convenience we put here с=1).
24
• As well known the given expression
determines a surface of hyperboloid on
pseudo-Euclidian momentum space called
a mass shell.
• Analogically we can assume that the
expression for the vector (J,σ) also gives
us a shell as some hyperboloid surface in
the pseudo-Euclidian space. It seems to
us that the presence of the invariant right
side says of some stability in respect of
such states inducing corresponding shell .
25
• Taking into account that using the
Bogoliubov transformations for QO we
come to thermal CCS now we could
connect this stability with some thermal
equilibrium at a conventional effective
temperature T*. That is why such a shell
could be called thermal “equilibrium shell”.
26
Effective action for QO in quantum thermostat
• Let us use the information obtain above for
behavior investigation of a quantum
oscillator in a thermostat. This model
allows us to calculate macroparameter J
making direct averaging of corresponding
operators because earlier in our
preceding paper we had obtained a wave
function for QO in thermostat.
27
It has the form
Here
Temperature T here fixes one of possible
thermal states i.e. plays a role of parameter.
28
Then one can calculate mean values of interest
and
It is self-understood that the equality
is ensured that is pleased..
29
Thus we recognize the results precisely
coinciding with the sequence predicted by QEST.
We only remind that J was
introduced there as a ratio between QO energy
and its frequency. We are of opinion that this fact
can serve as a good argument in support of our
effective action definition .
So, the concept of effective action is universal
one. We emphasis it is not connected with any
concrete object. By the example of QO we have
got confirmations that the given concept also has
a microscopic base.
30
Effective action and thermodynamic Lows
• Now let us return to the thermodynamics
language. In our opinion with a fair degree of
confidence we can extend the employment
region of this quantity. We suppose that
interconnection between effective action and
effective temperature T* having place for QO in
the form
still stand for any microobject in thermal
equilibrium. But we emphasis here one can
calculate J as mean value of operator
..
31
Thus on the base of the proportionality
starting from this moment we will write the
Zeroth Low of thermodynamics, as it
seems to us, in the more general form
• In so doing we translate the fluctuations
idea to the effective action. Here J and ΔJ
are the effective action and its fluctuation
of an object in thermal equilibrium but Jeq.
is the effective action of a quantum
thermostat.
32
• In other words we claim that in generalized
thermal equilibrium mean effective actions of
two contacting objects (of a microobject and a
quantum thermostat) coincide.
• Using imaginary we repeat the well known
aphorism “action is equal to contra-action” in its
literal sense. We are of opinion that such a way
of thermal equilibrium describing is more
adequate to nature than operating not only with
effective T* but even with customary (however
rather abstract) notion as temperature T.
33
• Going consecutively on this way we can rewrite
all thermodynamics formulas of QEST through
the quantity J including the canonical
distribution function in the macroparameters
space as well as the First and the Second Lows.
• Having the all necessary formulas one can also
obtain the expression for the effective entropy
34
We remark following:
• -it resembles in appearance to the formula
written above (in the beginning of the talk). But
there is a very essential difference by the sense
of the quantity J that here is the mean value of
the effective action operator;
• -now it has more wide sense for it can be
associated with any microobjects but not only
with QO;
• - according to it the minimal value of effective
entropy S*min. for any object is k B as before
35
On the Third Low of
Thermodynamics
• The last fact demands some additional
discussion that is quite appropriate here
because we have the proper instrument
for dispute in hands. It is useful to remind
that there are statements in literature
according to them a minimal value of
some “quantum” entropy Squ. can be
equal to non-zero but does not equal to
kB.
36
• An usual approach to calculating of minimal
quantum entropy Squ min. bases on the formula
• Here
and
amplitudes squared of
wave functions in basic state in q- and prepresentations.
• But we are of opinion that it is unsuitable for this
aim for under the sign of logarithm it has
dimensional terms.
37
• So we suggest to introduce new
dimensionless variables
where
is an arbitrary numeral constant.
Then the new dimensionless amplitudes
squared of wave functions are
38
• After that we get the expression for
minimal quantum entropy in the form
• If we substitute in it the wave function for
quantum oscillator in the basic state and
its Fourier-image we obtain the minimal
quantum entropy
• This result means that the answer depends on
the factor
39
We can consider the three variants as it is shown
in the table:
40
• Thus the result is determined by a
constant
choosing and in the frame of
purely quantum dynamics we can not get
the single-value one. But this choose we
can make single-value if we compare the
two minimal values: of the effective
and of the quantum
. It is easily to
see that they coincide at
giving us
41
We are pleased getting one and the same
result either from micro- and macrotheories. In our opinion, this fact is of great
importance:
• On one hand we have made some
contribution in the solving of the problem
with the constant in the Nernst theorem;
• On other hand we have confirmed the
significance of Boltzmann constant – it
gives us the minimal value of entropy.
42
Conclusion
• As a conclusion we allow ourselves to make
some remarks going out of the frame of the
given concrete results.
• We think that the idea of the operator
effective action introduction is fruitful
because it gives a chance to describe
objects moving nonperiodically.
• The agreement between thermodynamic (in
the frame of QEST) and quantum (in the
frame of TFD) approaches says, in our
opinion, that micro- and macrodescriptions
on the right footing can be used for the
solving of problems with quantum-thermal
phenomena simultaneously.
43
• Our proposed theory additionally elevates
the constant kB status from usual
dimensional factor to the level of the
fundamental constant of macrodescription.
• At the same time we emphasis that the
quantity
is also meaningful as the
fundamental constant of microdescription.
44
• From theoretical point of view, it would be
very interesting to establish interrelation
between the notions of spin projection
and effective action as the physical
characteristics connected with rotation
either in the Euclidian or pseudo-Euclidian
plane.
• It would be good to obtain some new
experimental confirms of fundamental
status of the two constants
and k В as
such as well as
joining the both
these quantities.
45
References
• O. Golubjeva, A. Sukhanov. Some geometrical
analogues between the describing of the states space
in non-classical Physics and the events space in
classical Physics. VI Intern. Conference BGL-08.
Debrecen, Hungary, 2008 (in print)
• A. Sukhanov. Some consequences of quantum
generalization of statistical thermodynamics . XIII
Intern. Conference “Problems of quantum field theory”.
Dubna, Russia, 2008 (in print)
• A. Sukhanov. A quantum generalization of
equilibrium statistical thermodynamics: effective
macroparameters. TMP, 154 (1), 2008
46
• A. Sukhanov. Schroedinger uncertainties relation
for a quantum oscillator in a thermostat. TMP,
148, 2 . 2006
• A. Sukhanov. Quantum oscillator in a thermostat
as a model thermodynamics of open quantum
systems. Physics of particles and nuclei. 36, 7a,
2005
• A. Sukhanov. Schroedinger uncertainties relation
and physical features of correlate-coherent states.
TMP, 132 (3), 2002
• A. Sukhanov. On the global interrelation between
quantum dynamics and thermodynamics. XI Int.
Conf. “Problems of quantum field theory”.
Dubna, Russia, 1999
47