( a) The dimensionality of f" cannot be smaller than 2 n X 2 n.

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(b) If {f,,} and {f;} are two sets of matrices satisfying (15.41), there exisb

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a nonsingular matrix S such that f" = Sf;S-I. The converse is obviously true. (c) Any 2 n X 2 n matrix is a linear combination of the unit matrix, the matrices f" (chosen to be 2 n X2 n ), and all the independent producb

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For n = 1, (15.41) defines two of the 2 X 2 Pauli spin matrices from which the third can be obtained as their product. It is obvious that any 2 X 2 matrix i~ a linear combination of the unit matrix and the Pauli spin matrices. For n = :. (15.41) defines the four 4 X 4 Dirac matrices Y,,' A possible representation of {f,,} by 2 n X 2 n matrices is

f 1 = ZI f3 f5

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An equally satisfactory representation is obtained by interchanging the roles of

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*These general properties are not necessary for future developments, since we work with ar. explicit representation. A general study of (15.41) was made by R.Brauer and H. Weyl, Am. J. Malh 57, 425 (1935).

THE ONSAGER SOLUTION

Xa and Z a (a = 1, ... , n). It is also obvious that given any representation, such as (15.43), an equally satisfactory representation is obtained by an arbitrary permutation of the numbering of f 1 , , f 2n It will presently be revealed that VI and V2 are matrices that transform one set of {f,,} into another equivalent set. Let a definite set {f,,} be given and let W be the 2n X 2n matrix describing a linear orthogonal transformation among the members of {f,,}:

f: = L w"vfv

(15.44)

where w"v are complex numbers satisfying

,,~1

w"v w,,>..

l)v>..

(15.45)

This may be written in matrix form as

(15.46)

where w is the transpose of w. If f" is regarded as a component of a vector in a 2n-dimensional space, then w induces a rotation in that space:

f{] [

[W ll

W 31

W2 WI 2n

(15.47)

2n 1

W 2n 2n

(15.48)

Substitution of (15.44) into (17.41) shows that the set {f:} also satisfies (15.41), because of (15.45). Therefore

f: = S(W)f"S-l(W)

where S( w) is a nonsingular 2 n X 2 n matrix. The existence of S( w) will be demonstrated by explicit construction. Thus there is a correspondence

S(w)

(15.49)

which establishes S( w) as a 2 n X 2 n matrix representation of a rotation in a 2n-dimensional space. Combining (15.48) and (15.44) we have

S(W)f"S-l(W) =

w"vfv

(15.50)

We call w a rotation and S( w) the spin representative of the rotation w. It is obvious that if WI and w2 are two rotations then W 1W 2 is also a rotation. Furthermore (15.51) We now study some special rotations wand their corresponding S( w). Consider a rotation in a two-dimensional plane of the 2n-dimensional space. A

SPECIAL TOPICS IN STATISTICAL MECHANICS

rotation in the plane JLV through the angle () is defined by the transformation.

f; f:

fA f" cos () - f v sin ()

("A=FJL,"A=Fv) (JL =F v) (JL =F v)

(15.521

f" sin () + fvcos ()

where () is a complex number. The rotation matrix, denoted by w(JLvl(}), explicitly given by

JLth column vth column

cos () - sin ()

sin () cos ()

JLth row vth row

(15.531

where the matrix elements not displayed are unity along the diagonal and zew everywhere else. Now w(jLvl(}) is called the plane rotation in the plane JLV. It l~ easily verified that

w(JLvl(}) = w(vJLI - (}) wT(JLvl(})w(JLvl(}) = 1

(15.541

The properties of wand S( w) that are relevant to the solution of the Ising model are summarized in the following lemmas. *

LEMMA 1

If w(JLv I())

S"v( (}), then S"v((})

= e-l/2or.r,

(15.55 I

Proof Since f"fv = - fvf" for JL =F v, (f"fv)2 = f"fvf"fv = -1. An identit\ analogous to (15.28) is

= cos - - f f sin 2 "V 2 Since (f"fv)(fvf,,) = (fvf")(f,,fv) = 1, we have

el/2orvr, e-l/2or.r,

e- 1/ 28r r ,

el/28rvr, e 1/ 28r,r.

e 1/ 20 (r.r,+r,r.)

(15.56)

Hence

"It will be noted that the proofs of these lemmas make use only of the general property (15.411 and the special representation (15.42) of {fp.}.

THE ONSAGER SOLUTION

A straightforward calculation shows that S".(O)fA S,,-;,l(O) = fA S". (0) f"S,,-;' 1( 0) S".(O)f.S,,-;,l(O)