NO'J'JOOO:O JAN MYRHEIM Nt:i-NO--4Hi Studies of Particle Statistics in One and Two Dimensions, Based on the Quantization Methods of Heisenberg, Schrddinger and Feynman ÆJSi DR. PHILOS. THESIS 1993 THE UNIVERSITY OF TRONDHEIM, AVH DEPARTMENT OF PHYSICS Fysisk institutt, Universitetet i Trondheim, AVH Studies of Particle Statistics in One and Two Dimensions, Based on the Quantization Methods of Heisenberg, Schrodinger and Feynman Jan Myrheim June 14, 1993 To Tora Acknowledgments This thesis contains five published articles, and I want to thank my collaborators and good friends Jon Magne Leinaas, Hans Hansson and Kåre Olaussen for their permission to include our joint work. Special thanks are due to Jon Magne who shares the responsibility for four of the five articles. Warm thanks go to my wife for much needed pushing. I want to thank the Research Council for Science and the Humanities. N'AVF. for the grant which enabled me to work on the last four articles, as well as the Theoretical Physics Group at the Institute of Physics, NTH, for their support, including a quiet place to work, with a faithful servant working day and night on long calculations, drawing figures, and printing beautiful manuscripts. Contents Acknowledgments i 1 Introduction 2 1.1 Quantum Mechanics or Identical Particles 2 1.2 Statistical Mechanics 5 1.3 Applications of Fractional Statistics 6 2 The Virial Expansion S 2.1 Bosons and Fermions 11 3 Summary of the Articles (I) to (V) 13 References 16 Articles I. J.M. Leinaas and J. Myrheim. Intermediate Statistics for Vortices in Superfluid Films. Physical Review B 37, 9286 (1988). II. J.M. Leinaas and J. Myrheim. Quantum Theories for Identical Particles. International Journal of Modern Physics B 5. 2573 (1991). III. J.M. Leinaas and J. Myrheim. Heisenberg Quantization for Systems of Identical Particles. International Journal of Modern Physics A 8. 3649 (1993). IV. T.H. Hansson, J.M. Leinaas and J. Myrheim. Dimensional Reduction in Anyon Systems. Nuclear Physics B 384. 559 (1992). V. J. Myrheim and K. Olaussen. The Third Virial Coefficient of Free Anyons. Physics Letters B 299. 267 (1993). erratum B 305, 428 (1993). 1 1 Introduction The theme of this thesis is the application of different methods to the quantization problem for systems of identical particles in one and two dimension*. It i» treated in five articles, listed above in the table of contents and referred to below a» (1) to (V). The standard method is the analytic quantization method due to Schrodinger. which leads to the concept of fractional statistics in one and two dimensions. Two-dimensional particles with fractional statistics are well known by the name of anyons. Two alternative quantization methods are shown here, the algebraic method of Heisenberg and the Feynman path integral method. The Feynman method is closely related to the Schrddjnger method, whereas the Heisenberg and Schrodinger methods may give different results. In article (I) the Heisenberg method is applied to the equations of motion of vortices in superftuid helium, which have the form of Hamiitoniau equations for a one-dimensional system. In (II) and (III) the same method is discussed more generally for systems of identical particles in one and two dimensions. The relation between the Heisenberg and Schrodinger methods is discussed in (IV). Finally, article (V) presents an application of the Feynman method to the problem of computing the equation of state for a gas of anyons. The remaining part of this introduction is a brief review of some of the back­ ground. Some more background material for article (V) is presented in Section 2. Section 3 gives a brief summary of the contents of the articles. 1.1 Quantum Mechanics of Identical Particles One of the many peculiarities of quantum mechanics is the fact that identical particles always interact, simply because they are identical. Immediately after Heisenberg and Schrodinger formulated quantum mechanics as it i* known today. Heisenberg and Dirac extended the theory to systems of identical particles [I -•'{]. Their key observation was that all operators representing observables in such sys­ tems have to be symmetric under any interchange of particle labels, if there is really no observable difference between the particles. This innocent looking state­ ment has deep consequences, because symmetric operators preserve the symmetry properties of the wave functions. For example, if the operator A and the wave function v are both totally symmetric, then the wave function Av is also totally symmetric. "++ = +". And similarly, if A is symmetric but r is totally antisym­ metric, then Av is totally antisymmetric. "+— = -". Consequently, there exists a complete quantum theory of identical particles using only the totally symmetric wave functions, and there exists a different complete theory using only the totally 2 antisymmetric wave functions. The symmetry or antisymmetry of the allowed wave function;» U d charac­ teristic property of a given system of identical particles, usually referred to a» the statistics of the particles. Particles described by symmetric wave functions satisfy Bose—Einstein statistics and ire called bosons. Particles described by antisymmetric wave functions satisfy Fermi—Dime statistics, they are/erniion». and because of the antisymmetry they obey the Pauli exclusion principle, that two particles can not occupy the same quantum state. The symmetry or antisym­ metry results in an effective attraction between bosons and an effective repulsion between fermions, both of a purely quantum mechanical nature. We may refer to this kind of attraction or repulsion, related to the particle statistics, as a statistics interaction. The mutual repulsion between fermions is quite literally a tangible fact: we can walk on the earth because matter consists of a small number of dif­ ferent species of fermions. In fact, the stability of matter (at least the proof of stability) depends on the fermionic nature of matter [4j. Since the theory of Heisenberg and Dirac predicted that identical particles had to be either bosons or fermions. and since this prediction was verified experimen­ tally, there was not much need for a better theory. However, the theory could be questioned on philosophical rather than experimental grounds. Consider, for ex­ ample, particles that are so far apart that they can not be physically interchanged. Then it is intuitively obvious, and indeed true, that it does not matter whether we symmetrize or antisymmctrize our wave functions, or do neither of the two. This example suggests that the symmetrization or antisymmetrization postulate is not truly fundamental, but is rather a consequence of some more fundamental principle. Furthermore, an interchange of identical particles is obviously an identity transformation from the physical point of view. In quantum mechanics it is not unusual that a physical identity transformation is represented mathematically by a phase factor, since two wave functions represent the same physical state if they differ only by an overall phase factor. Any permutation of bosons is represented by the trivial phase factor +1. whereas even and odd permutations of fermions are represented by +1 and —1. respectively. A natural question is then, why only ±1 and not more general complex phase factors? Laidlaw and DeVV'itt answered this question when they applied the r'eynman path integral formalism to systems of identical particles [5]. In this formalism the interchange of identical particles has a clear physical meaning as a continuous process in which each particle moves along a continuous path. The path depen­ dence of the interchange is all important, since it relates the quantum mechanical concept of particle statistics to the topology of the classical configuration space. 3 The phase factors associated with different interchange pat lis must dc-finr a rvfin- sentation of the first homotopy gioup (the fundamental group) of the configuration space [6|. This requirement leads to the conclusion thai only bosons and ferniious can exist in Euclidean space of dimension three or higher, whereas more general possibilities open up in the two-dimensional case. The formalism does not apply in one dimension. Using a more traditional approach to quantization. Jon Magne Leinaas and I derived the same relation between particle statistics and topology |7]. Our approach was based on the geometrical interpretation of wave functions which is the basis of gauge theories, and which goes back to Weyl and Dirac [8. 9|. We studied in some detail the more general kinds of statistics allowed in one- and two- dimensional systems. In either case there exists a continuously variable parameter defining the statistics, interpolating continuously between Bose—Einstein and Fermi—Dirac statistics. In one dimension the parameter may be interpreted as the strength of a zero range potential between bosons, and when the strength becomes infinite, the bosons become fermions. This particular one-dimensional potential, the Dirac ^-function, is one of the few examples where the many-body problem is exactly soluble [10-13]. In two dimensions the parameter may be chosen as a phase angle which is 0 for bosons and r for fermions. and we showed by the example of the two-dimensional harmonic oscillator that the continuous variation of the phase angle gives a continuous interpolation between the boson and fermion energy spectra. The intermediate statistics, as we called it. is now usually called fractional statistics. In the two-dimensional