*424*
*118*
*3MB*

*English*
*Pages [177]*
*Year 2005*

*Table of contents : 1 GENERALIZED STATISTICS OF INDISTIGUISHABLE PARTI-CLES 11.1 The Symmetrization Postulate 11.2 Identicality and Indistinguishability 21.3 The Quantum Statistics of Indistinguishable Particles 51.4 Cluster Decomposition: Systems and Subsystems 71.5 Conclusion 82 STATISTICAL MECHANICS OF IDENTICAL PARTICLES 92.1 Multiparticle States and the Permutation Group 92.1.1 Notation and Fimdamentals 102.1.2 Frobenius Symbols 102.2 Partition Fvmctions for (In)distinguishable Particles 122.2.1 Parabosons and Parafermions 192.2.2 Beyond Parabosons and Parafermions 282.3 Degenerate Systems 352.4 Conclusion 373 STATISTICAL MECHANICS OF EXTENSIVE SYSTEMS 393.1 Extensivity 393.2 Extensivity and Positive Counting 413.3 Classification 453.4 Symmetry Types in Fuzzy Statistics 473.5 Examples and Comparisons 503.5.1 Interpolating Between Bosons and Fermions 503.5.2 Haldane/Wu and Polychronakos's Fractional Statistics 523.5.3 Gentile Statistics 543.6 Conclusion 564 SECOND QUANTIZATION AND GENERALIZED STATISTICS 574.1 Algebras: Bilinear and Trilinear Relations 584.2 Actions of the Permutation Groups 634.2.1 Gram Matrices and the Representations of the PermutationGroup 674.2.2 Clustering Revisited 814.3 Classification 854.3.1 Examples 904.4 Known Models 954.4.1 Green's Parastatistics 954.4.2 Greenberg's Quons 964.4.3 Meljanac's proposal and Govorkov's Parastatistics 964.5 Conclusion 985 ENSCRIPTION AND TRANSLATION OF QUANTUM TEXTS . 1005.1 Cloning and Quantum Information 1005.2 Enscription 1045.3 Enscribing 2-texts 1135.4 Enscribing Real-Uniform N-texts 1165.4.1 Real-Uniform 3-Texts 1195.4.2 Real-Uniform N-Texts 1285.5 The Structure of an Enscribable Text 1335.6 Enscription as a Resource: Probabilistic Clorung 1355.7 Translation 1425.8 Graphs and the Structure of Translatable Texts 145CITED LITERATURE 162*

QUANTUM COUNTING AND WRITING: EXOTIC QUANTUM STATISTICS AND THE TRANSLATION OF QUANTUM TEXTS

BY RANDALL ESPINOZA B.S. Universidad de Costa Rica 1994

THESIS Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics in the Graduate College of the University of Illinois at Chicago, 2005

Chicago, Illinois

UMI Number: 3199854

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THE UNIVERSITY OF ILLINOIS AT CHICAGO Graduate College CERTIFICATE OF APPROVAL

1 hereby recommend that the thesis prepared under my supervision by RANDALL ESPINOZA

entitled

QUANTUM COUNTING AND WRITING: EXOTIC QUANTUM STATISTICS AND THE TRANSLATION OF QUANTUM TEXTS

be accepted in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY

cr-yy1 is denoted by J and

labeled by the arrays {jr+i, ••• ,jM), with ;V+i < • • • < /m- Expanding the determinant in the numerator of

(x, y) we have

.v;x)=^E='8nw ^ > O-

n C"' ,V

n i=jr+1,-,iM

22

It should be clear from Equation 2.21 that function for the partition

. .,V;x) equals, up to a sign, the Schur

defined by the array

(I) _ \ M + q+i k - l , i k i n l ,

' M - jk,

h > • • • > ir)

jk in J ,

jr+i < • • • < ; m -

The partition ?J clearly has a diagonal of length r (which is how many

are bigger than

M) and is best represented by its Frobenius symbol

Fr(Ai) = [ q + h - l , . . . , q + i r - l \ h - l , . . . , i r - l ] .

(2.36)

(z'L, ..., zV; x) and the Schur function S AJ (X ) is the result of the

The relative sign between

permutation of the lower indices in Equation 2.34 for

{h,---, ir'> x) relative to the order of

the columns in the determinant in Equation 2.21 — that is, it is the sign of the permutation

and is given by

H

^2

• ••

V

1

2 ...

r

]r+l

• • •

r + 1 ...

]M

\

M )'

'

Using the form of Fr(Ai), it follows that

E(^'-l) = |bAzllei We conclude that

/n oyx

^

(2-38)

..., zV) = (-1)I'''^ISAJ (X ), and more generally that/(°'^^(x,y) is given

by det(x^-' + vx''.^ ' , M-,x

)

)i 0, or of the form (i, i + |sl) — the diagonal starting at the th box of its first column — for s < 0. We denote by rA(s) the length of the s-diagonal of A. It follows that for a partition A = (Ai,..., Am), the length of the s-diagonal rx{s) is given by the number of parts Ayt that satisfy

> k + s. Equivalently, in terms of the associated array

i = {l\,.. .,£m) introduced in Equation 211, the s-diagonal is the number of 4's for which 4>M

-I-

s. Figure 2.2.2 clarifies this concept with some examples.

33

Fr(X) = [3,1,014,2,1]

Fr(\) = [4,2/ll3,1,0]

1-diagonal (- I )-diagonaI '-x(-l) = 3 Figure 2, Two examples of the concept of the s-diagonal.

Defiiie the function ^s(0 on the ring of formal symmetric fimctions as

(2.81) A

Since ^-s(0 = ^s(0/ we assume without loss of generality that s > 0. When dealing with the CGPF of systems with (p,£^)-statistics, the parameter s will usually stand for p - q. For instance, for p - q > 0 the (p, (j)-envelope

introduced before can be equivalently defined

as the set of all those partitions A for which r,\ (p - q) < q. With this in mind it follows that Lemma 2.7. f4^ = EZ(,(2-82)

34

Proof. The left hand side of Equation 2.82 is

( £ i") A

/

\n=0

/

A

(2.83)

«=rA(s)

00

'?=0

E «a(x )), A,rA(s)