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Decoherence of colour in QCD vacuumDecoherence of colour in QCD vacuumDecoherence of colour in QCD vacuumDecoherence of colour in QCD vacuum

V.I. Kuvshinov V.I. Kuvshinov

P.V. BuividovichP.V. Buividovich

Joint Institute for Power and Nuclear Research – SosnyJoint Institute for Power and Nuclear Research – Sosny

X International School-SeminarX International School-Seminar

July 17July 17, 2009, 2009

GomelGomel

BELARUS BELARUS

The model of QCD stochastic vacuumThe model of QCD stochastic vacuumThe model of QCD stochastic vacuumThe model of QCD stochastic vacuum

The model of QCD stochastic vacuum is one of the popular The model of QCD stochastic vacuum is one of the popular phenomenological models which explains quark phenomenological models which explains quark confinementconfinement [Sawidy'77, Ambjorn'80, Simonov'96, Dosch'02, [Sawidy'77, Ambjorn'80, Simonov'96, Dosch'02, Simonov'04]Simonov'04]

It is based on the assumption that one can calculate vacuum It is based on the assumption that one can calculate vacuum expectation values of gauge-invariant quantities as expectation values of gauge-invariant quantities as expectation values with respect to some well-behaved expectation values with respect to some well-behaved stochastic gauge fieldstochastic gauge field

It is known that such vacuum provides confining properties, It is known that such vacuum provides confining properties, giving rise to QCD strings with constant tension at large giving rise to QCD strings with constant tension at large distancesdistances

White mixtures of statesWhite mixtures of states White mixtures of statesWhite mixtures of states

Most frequently the model of QCD stochastic vacuum is used Most frequently the model of QCD stochastic vacuum is used to calculate Wilson loops, string tensions and field to calculate Wilson loops, string tensions and field configurations around static charges configurations around static charges [Simonov'96, Dosch'02][Simonov'96, Dosch'02]

In this paper we will consider the colour states of quarks In this paper we will consider the colour states of quarks themselvesthemselves

Usually white wave functions of hadrons are constructed as Usually white wave functions of hadrons are constructed as gauge-invariant superpositions of quark colour statesgauge-invariant superpositions of quark colour states

Here we will show that white objects can be also obtained as Here we will show that white objects can be also obtained as white mixtures of states described by the density matrixwhite mixtures of states described by the density matrix.. Suppose that we have some quantum system which interacts Suppose that we have some quantum system which interacts with the environmentwith the environment

QCD stochastic vacuum as eQCD stochastic vacuum as environmentQCD stochastic vacuum as eQCD stochastic vacuum as environment

Suppose that we have some quantum system which interacts with the Suppose that we have some quantum system which interacts with the environmentenvironmentInteractions with the environment can be effectively represented by additional Interactions with the environment can be effectively represented by additional stochastic terms in the hamiltonian of the systemstochastic terms in the hamiltonian of the systemThe density matrix of the system in this case is obtained by averaging with The density matrix of the system in this case is obtained by averaging with respect to these stochastic termsrespect to these stochastic terms [Haken'72, Reineker'82, Haake'91, Peres,95][Haken'72, Reineker'82, Haake'91, Peres,95]

QCD stochastic vacuum can be considered as the environment in quantum-QCD stochastic vacuum can be considered as the environment in quantum-optical languageoptical languageInstead of considering nonperturbative dynamics ofInstead of considering nonperturbative dynamics ofYang-Mills fields one introduces external stochastic field and averageYang-Mills fields one introduces external stochastic field and averageover its implementationsover its implementations [Savvidy'77, Ambjorn'80, Simonov'96, Dosch'02, [Savvidy'77, Ambjorn'80, Simonov'96, Dosch'02, Simonov'04]Simonov'04]

Interactions with the environment result in decoherenceInteractions with the environment result in decoherence and and relaxationrelaxationof quantum superpositions of quantum superpositions [Haake'91, Peres,95][Haake'91, Peres,95]

Information on the initial state of the quantum system is lost afterInformation on the initial state of the quantum system is lost aftersufficiently large time. Here the analogy between QCD vacuum and thesufficiently large time. Here the analogy between QCD vacuum and theenvironment can be continued: environment can be continued: information on colour states is also lostinformation on colour states is also lostin QCD vacuum due to confinement phenomenonin QCD vacuum due to confinement phenomenon

Colour density matrixColour density matrixColour density matrixColour density matrixTo demonstrate the emergence of white states which is caused by To demonstrate the emergence of white states which is caused by decoherence processes consider propagation of heavy spinless decoherence processes consider propagation of heavy spinless quark along some fixed path γ from the point quark along some fixed path γ from the point x x to the point to the point yy. The . The amplitude is obtained by parallel transportamplitude is obtained by parallel transport

kets are colour state vectors, kets are colour state vectors, is the path-ordering operator and is the path-ordering operator and AμAμ is the gauge field vector. Equivalently we can describe is the gauge field vector. Equivalently we can describe evolution of colour state vectors by parallel transport equation evolution of colour state vectors by parallel transport equation

In order to consider mixed states we introduce the In order to consider mixed states we introduce the colour density colour density

wk wk is the probability to find the system in the state is the probability to find the system in the state

Colour density matrix evolutionColour density matrix evolutionColour density matrix evolutionColour density matrix evolutionWe first obtain the colour density matrix of the quark which We first obtain the colour density matrix of the quark which propagates in a fixed external gauge field, which is some propagates in a fixed external gauge field, which is some particular implementation of QCD stochastic vacuum. We particular implementation of QCD stochastic vacuum. We will denote this solution by . The colour density will denote this solution by . The colour density matrix is parallel transported according to the matrix is parallel transported according to the following equation: following equation:

In order to find the solution of this equation we decompose In order to find the solution of this equation we decompose the colour density matrix into the pieces which transform the colour density matrix into the pieces which transform under trivial and adjoint representations of the gauge group:under trivial and adjoint representations of the gauge group:

Averaging over stochastic gauge fieldAveraging over stochastic gauge field Averaging over stochastic gauge fieldAveraging over stochastic gauge field

According to the definition of the density matrix we should finally average According to the definition of the density matrix we should finally average this result over all implementations of stochastic gauge field. this result over all implementations of stochastic gauge field. In the model of QCD stochastic vacuum only expectation values of path-In the model of QCD stochastic vacuum only expectation values of path-ordered exponents over closed paths are defined.ordered exponents over closed paths are defined. Closed path corresponds to a process in which the particle-antiparticle Closed path corresponds to a process in which the particle-antiparticle pair is created, propagate and finally annihilate pair is created, propagate and finally annihilate Due to the Schur’s lemma in colour-neutral stochastic vacuum it is Due to the Schur’s lemma in colour-neutral stochastic vacuum it is proportional to the identity, therefore we can write it as followsproportional to the identity, therefore we can write it as follows

where by we denote averaging over where by we denote averaging over implementations of stochastic vacuum and implementations of stochastic vacuum and is the Wilson loop in the adjoint representationis the Wilson loop in the adjoint representation

Colour density matrix formColour density matrix formColour density matrix formColour density matrix form After averaging over implementations of stochastic vacuum After averaging over implementations of stochastic vacuum we obtain for the colour density matrix of the colour charge we obtain for the colour density matrix of the colour charge which was parallel transported along the loop γ:which was parallel transported along the loop γ:

This expression shows that if the Wilson loop in the adjoint This expression shows that if the Wilson loop in the adjoint representation decays, the colour density matrix obtained as representation decays, the colour density matrix obtained as a result of parallel transport along the loop γ tends to white a result of parallel transport along the loop γ tends to white colourless mixture with colourless mixture with

where all colour states are mixed with equal probabilities where all colour states are mixed with equal probabilities and all information on the initial colour state is lostand all information on the initial colour state is lost

Colour density matrix and Wilson area lawColour density matrix and Wilson area law Colour density matrix and Wilson area lawColour density matrix and Wilson area law

Wilson loop decay points at confinement of colour charges, therefore Wilson loop decay points at confinement of colour charges, therefore the the stronger are the colour charges confined, the quicker their states stronger are the colour charges confined, the quicker their states transform into white mixturestransform into white mixtures. It is important that the path γ is closed, . It is important that the path γ is closed, which means that actually one observes particle and antiparticlewhich means that actually one observes particle and antiparticle

As the Wilson area law typically holds for the Wilson loop, we can obtain As the Wilson area law typically holds for the Wilson loop, we can obtain an explicit expression for the density matrix. Here it is convenient to an explicit expression for the density matrix. Here it is convenient to choose the rectangular loop γ R×T which stretches time T and distance choose the rectangular loop γ R×T which stretches time T and distance R:R:

where is the string tension between charges in the where is the string tension between charges in the adjoint representation, is the string tension between charges in the adjoint representation, is the string tension between charges in the fundamental representation and are the eigenvalues of fundamental representation and are the eigenvalues of quadratic Casimir operators. Here we have used the Casimir scaling quadratic Casimir operators. Here we have used the Casimir scaling [Simonov’96, Dosch’02, Simonov’04][Simonov’96, Dosch’02, Simonov’04]

PurityPurity

PurityPurity

We can obtain the decoherence rate, which is introducedWe can obtain the decoherence rate, which is introducedusing the concept of purity . For pure states the using the concept of purity . For pure states the purity is equal to one purity is equal to one For our colour density matrix the purity is For our colour density matrix the purity is

Purity decay rate is proportional to the string tension and the Purity decay rate is proportional to the string tension and the distance distance R.R.

Purity decay rate is proportional to the string tension and the Purity decay rate is proportional to the string tension and the distance distance R. R. It can be inferred from this expression that It can be inferred from this expression that the the stronger is the quark-anti quark pair coupled by QCD string stronger is the quark-anti quark pair coupled by QCD string or the larger is the distance between quark and anti quark, or the larger is the distance between quark and anti quark, the quicker information about colour states is lost as a result the quicker information about colour states is lost as a result of interactions with the stochastic vacuumof interactions with the stochastic vacuum

ConclusionsConclusions ConclusionsConclusionsWe show that in QCD stochastic vacuum white states of We show that in QCD stochastic vacuum white states of colour charges in the fundamental representation of colour charges in the fundamental representation of SU(NC) SU(NC) gauge group can be obtained as a result of decoherence of gauge group can be obtained as a result of decoherence of pure colour state into a mixed statepure colour state into a mixed state

Decoherence rate is found to be proportional to the tension Decoherence rate is found to be proportional to the tension of QCD string and the distance between colour charges.of QCD string and the distance between colour charges.

The purity of colour states is calculatedThe purity of colour states is calculated

Remark: There exist direct connections among decoherence, Remark: There exist direct connections among decoherence, colour loosing, purity evolution, confinement,colour loosing, purity evolution, confinement,

and also and also [Kuvshinov, Kuzmin, Buividovich (2005)][Kuvshinov, Kuzmin, Buividovich (2005)] entanglement, chaotic colour behaviour, fidelity in QCD entanglement, chaotic colour behaviour, fidelity in QCD vacuum vacuum

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ReferencesReferencesReferencesReferences

Thank you for the attention!Thank you for the attention!