sistemi relativisticitheory.fi.infn.it/becattini/files/lezione5.pdf · the operators are defined as...
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Sistemi relativistici
Corso di laurea magistrale 201415
F. Becattini
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Sommario lezione
●Insieme microcanonico completo per un sistema relativistico
●Gas ideale quantorelativistico con momento angolare intrinsico fissato
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Most general microcanonical ensembleMicrocanonical =
ensemble of states of an isolated system, with all conserved quantities fixed
Pi must be a projector onto ALL CONSERVED QUANTITIES. But can we conserve
energymomentum and angular momentum at the same time? Not in Quantum Mechanics
We have seen that for nonabelian groups this means projecting onto an irreducible vectorof the full symmetry group
The full symmetry group for spacetime is the orthocronous Poincare' group
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Construction of irreducible statesof Poincare' group (1)
We use the socalled method of induced representations and consider just the massive case.
First observe that fourmomenta are commuting generators, therefore we can constructstates like
In order to identify the missing degeneracy indices, one notes that a given fourmomentum is invariant under some subgroup of Lorentz transformations, defined as little group.
For massive particles, the little group is a group of rotations in the restframe of the particle,i.e. the frame where P = (M,0) (we do not consider massless case)
Therefore, the little group is momentumdependent.
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Construction of irreducible statesof Poincare' group (2)
Define PauliLubanski vector
fulfilling these commutation relations
Decompose it along 3 spacelike axes forming an orthonormal frame with P
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The operators
are defined as the spin operators and form an SU(2) Lie algebra
is a Casimir of this algebra and it is also a Casimir of the full Poincare' algebra.
S3 and S2 can be diagonalized along with P (eigenvalues λ and J(J+1))
NOTE choosing
the eigenvalue of S3 is the component of the angular momentum in the
rest frame of the particle along the direction of particle momentum (helicity)(EXERCISE)
is the parity
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Full projector for IO(1,3)↑
including a parity quantum number
Generalize integral expression
BEWARE Strictly speaking projectors can only be used for compact groups. The above expression is not idempotent (in fact P2 = c P with c infinite)Yet, the same happen with the “projector”
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General decomposition of Poincare' transformations
pure boost rotationI or translation
the exponent z=0 for identity and =1 for the reflection
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Work in the rest frame, where P=(M,0)
vanishes unless gz is a pure rotation or a reflection because a boost
changes the energy
Achieved effective reduction to
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Final expression
provided that P=(M,0)
We have achieved factorization of energymomentum and spin projectors in the rest frame
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Ideal gas of particles with spin
Boltzmann statistics: straightforward extension of the single particle case
Quantum statistics: more complicated
Henceforth, we will confine ourselves to the Boltzmann case
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Special case: sphere
Take advantage of the rotational invariance of the space integrals to reducegroup integration dimension
The proof is similar to the one for inner symmetries. If [P
V, R]=0 (spherical symmetry), the partition function is independent of
the third component λ and we can replace the matrix element with the trace
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Grandcanonical limitGrandcanonical partition function (for V large and J finite)
If also J is large, then
and
which corresponds to a classical limit were not for the presence of the traces
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Classical limit
It is obtained by reintroducing in the formulae and taking the limit After some manipulations, one gets:
This is reasonable: in the limit of classical mechanics, angular momenta are commutingconserved vectors
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Grandcanonical partition function with fixed angular momentum – J large
Saddlepoint expansion for J and V large: introduction of a rotational potential (=angular velocity)
L
S
Macroscopic rotating systems:GCE with large J
J