Sistemi relativistici

Transcription

1 Sistemi relativistici Corso di laurea magistrale F. Becattini

2 Sommario lezione Problemi dei sistemi piccoli Trattazione generale dell'insieme microcanonico per un campo libero

3 Problems The large volume hypothesis has been tacitly introduced in microcanonical and canonical calculations when writing In all our derivations we have written the four momentum constraint but four momentum is NOT well defined in a finite volume! (see e.g. Landau, 3 rd volume) Even for an ideal gas, things become quite difficult in a proper quantum field theoretical framework. Thermal field theory has been developed for large systems, but not for small systems Several corrections due to finite volume should be expected when E 3 V~ 1 (no mass scale in the problem) or m L ~ 1

4 Re starting from scratch... What does finite volume mean? Simplest case: one free particle in NRQM. Finite volume: infinite potential wall. Everybody knows the solution... The microcanonical partition function (traditional definition): but this function, unlike in the classical case, is a delta like and not continuous function of E

5 A better definition The MPF is the trace over localized states. Localized states are NOT a basis of the full Hilbert space, only of the Hilbert space corresponding to the finite region.

6 One particle in NRQM reexamined Therefore, using the new definition and inserting a complete set of momentum eigenstates: So:

7 Because of the completeness relation in the box: Therefore: This is a continuous and derivable function of E! This is the same expression as in classical mechanics, for a single free particle confined within a box with volume V

8 MPF for a relativistic system Inserting a complete set of states, one writes it as a trace over the full Hilbert space: where In principle, the relativistic problem should be tackled within a quantum field theoretical framework

9 Remarks 1. For the calculation of mean values, one should be very careful if The density operator MUST be hermitian, therefore a better definition would be: The MPF is the same as before because of the ciclicity of the trace and now mean values are unambiguous. Another possibility is: 2.The same problem may occur for the canonical ensemble: in this case the former re definition Is obviously better

10 Difficulties in quantum field theory The above expression can be quite easily worked out for an ideal gas in a first quantization framework, where the number of particles is an observable which does not depend on whether particles are confined or not free state confined However, in QFT this no longer holds. Instead kinematical variables particle multiplicities NOTE: Henceforth I will focus on a neutral scalar (spin 0) field

11 Detector The detector can count photons even if the box is empty (= in its vacuum state) Related to the Casimir effect

12 We have to compare a state of the field in a box with a state of the field in the whole space. Not easy, because they are in principle different problems IDEA: Map the localized Hilbert space H V onto the full Hilbert state H Now the problem can be solved Bogoliubov relations for the creation and destruction operators of a spinless boson field EXERCISE: Derive it k are discrete vectors

13 In principle, the Bogoliubov relations could be used to work out the microcanonical partition function. This is, however, a very laborious method ALTERNATIVE FUNCTIONAL METHOD Subtlety: the state of the field includes its degrees of freedom over the whole space, otherwise some of the scalar products involving asymptotic states in the following derivation could not be calculated. However, the final result must be independent of the field states outside V

14 Le prossime 5 trasparenze possono essere saltate, solo se siete interessati Working out Obtaining the final expression involves several steps and lenghty calculations. 1. Expand Ω in a series of fixed multiplicities N 2. Use a basis of momentum eigenstates and focus on the single particle term with N=1

15 using the relations where 3. We are led to the calculation of a (gaussian) functional integral See e.g. Weinberg QFT vol. 1, p. 387 (Without P V )

16 Result: where the inverse kernel K 1 V fulfills being is a positive constant that is left unexpanded

17 4. Complete the calculation by getting rid of spurious degrees of freedom This the same result as in NRQM modulo the last factor!

18 5. Extend to the case of multiparticle states, by taking into account the following expression After some work, one gets to: where r is a permutation and the Fourier integrals appear

19 Final expression of MPF The Fourier integrals express the so called Bose Einstein correlations: bosons from a finite volume are closer in momenta than distinguishable particles. For large V: and the terms corresponding to permutations give rise to the clusters of identical particles that we derived by enforcing momentum conservation writing that is:

20 Remarks Similar expression can be found for charged bosons The expression of the Ω is the same as it would have been obtained in a 1 st quantization framework except the immaterial factor This means that summing over all localized states cancels out the effect of Bogoliubov transformations, i.e. the effect of the difference between localized and asymptotic states