6. ANÁLISIS INTERNO
6.3 FORTALEZAS Y DEBILIDADES (FS – DS)
6.3.2 FORTALEZAS Y DEBILIDADES
Expand Kn — — x\n — — where ^Kn = ^ > l C x )
We can now use the Theorem to evaluate the Hamiltonian of our Model Field II, As given in Appendix 7.3 the result is
oj H = QL^ ^ !l ! C4n ^Kn'^B %n) Ti - Q - n,K%0 00 — — — -j- ®n 2 ^ gCig®k * 4 ^ - T i 4 „ 1 = '!= • * # ( 4 4 - gOo j / " n % -
+ higher terms, where g^*^ is the metric tensor for the 4-surface. g^"^ and may be expressed in terms of C » and tt as seen in
> 1
Appendix 7.3 and > Note that <€.>_, <%>_,
n Ç, j "" ,I ] n j I* “ - -“
IT = <n> . n *'
depend solely on and p^^ Now define
137. 'n *^n "'n Kn rc-g„) 4ft C-g^) \ '^4A^Cg°ig°4 gfg°°)Kj4j - - ’ wKn
{<'Ç-
4'"'
g.n00where repeated indices imply a summation. Then H = QL^ S a/(1+(V53„)(1+ ^ < u > h n 5 (T - + ^'"Kn 4 n \ n + hasher terms ”n (7.5-1)
To quantize it, the canonical variables Ç,Tr are to be regarded as operators in a linear vector space with the standard equal-time commutation rules
[Ç(x),Tt(x)] =
i-fi
5(x-x) ; [Ç(x),Ç(x)] = [x(x),n(x)] =o
These imply the commutation rules for q^^, p^^ t
The everywhere-slowly-varying nature of the field ^ means that may be taken as unquantized c-numbers (see Appendix 7.4 for details). As a result, g^*^ may also be similarly treated as c-numbers
138-
•j*
To this approximation, we see that a^^^ a^^ defined above obey the standard commutation rules for creation and annihilation operators, i.e.
’ ^*Kn'*Kn] ° ° °
Indeed with the above expression for the Hamiltonian, we can regard
4*
*Kn'^Kn ^^spectively as the annihilation and creation operators for the quantum of energy-hw^^.
Note that if we write
%Kn) = ''Kn/': and (k^^) _ = - --I>'^rJ ,
where are the three components of K, (i = 1,2,3), then
This means that
(W^„/C , -K3 .
are the covariant components of a null vector with respect to the
^ . TK
metric g^ .
Now the situation is that we have on the one hand the slowly-varying classical field g^ and on the other hand the quantized field with the Hamiltonian (7,5-1). The background metric g^*^ depends on the
classical variables <Ç > and ,] n <-~yr>Q n so that it will vary slowly with^ ^ time as these variables develop in time according to the classical
139
canonical equations generated by the Hamiltonian QL^ SV[1+<VÇ4] [l+<^>2] .
n - -
i"
The quanta created by a^^ from the vacuum state are mainly in the
i i i
region n - L/2 < x < n + L/2, but there is some overlap with neighbouring regions. The total number operator is
N = 2 ,
n
where N = S a^ a^ n ^ Kn Kn and [N ,N /] = 0 .^ n' n^
It is seen that we have essentially a free field theory. To bring in interaction, one may proceed to higher order terms in the Taylor series. However the calculation becomes very lengthy and the usual divergent problem remains.
140- 7.6 Appendix 7.1 Properties of (A) Fourier Transform
/2ïï (x) -ikx T l " -iknL . ^ ^ tt . , „ tt .. e = — e 1 L dx e 2TT if K - T- < k < K +
é
k l ~ if k = K ± ^ = 0 otherwise (B) Orthogonality (C) Completeness = Ô(x-x) (D) 3UKn m-n 9x ^Kn * ? 'L(m-n) m^n u."Km * * & d x Case I : K - K = 0 pnn 4/L3__ if p = n = n 2tt /h (p-n) if p = n ^ n (-1) 2tt^ /L p+n+n r (-l)P . C-D" + (-1)" -, Cp-n) Cp-n) (n-pTfn-ri') (n-p)Xn-n)'^ ’ if p ^ n ^ n^141. f 2 TT Case IX: K - K = ^ ,2w/L 1 . . f I / = if p = n = n P"" 8/L = if p = n , / 4tt ÆCp-n) 4Tr/L(p-n) = . M il'"'" r (-l)P . + M ) " + (-1)"' , 4tt^/L Cp“n)(p-n) (n-p) (n-X) (n-p) Case III:
^pni/ ” ^ ail other cases apart from those obtainable by K— K
the symmetry properties the expression I K-K
Symmetry Properties of 1^^^/ ^ ^ pnn/ ~ pnn'
(b) The order of the indices pnn^ is irrelevant to the value of pnn'’
X - K _K-K ^ ,
Ipnn = V p = ^npn ^"d so on, Ipp^/ = ^4pp
Our present set of functions may have certain similarities with the Block functions expressed in terms of the Wannier functions in solid
G1
-142- 7.7 Appendix 7.2 The Theorem
2
I . d. p w . w ) . 4 . t(|ê£jër) Î « L » b J
n ^ a D n K^O /
Proof :
Expanding the number 1 in terms of the complete orthonormal set we obtain
1 = Æ s
n so I = / dx F(e a (x))= Æ S / d \ U F(on e ) = s a In I m = Æ / dx u on ^ aF(e )We assume 0^ to be slowly varying functions of x so that the main contribution to I comes from the values of 0 near x = nL in whichn a n region 0^ = 0 (x) - <0 > « 1.^ an a a n
Let F(0^) = ^ ®an^’ then a Taylor expansion gives
4 V = " f ^ « a V + (Ir) < n ' a n \ a b^n 'an ^ n ' ...
-143.
Now our assumption that depends mainly on the values of 0^ near = nL shall mean that the same applies to each integral over a Taylor series term. Let 8^^(x) be of the order of, say A << 1,
for y near x = nL, then we can roughly estimate that / dx ^ n " ^ ^ U 0^ 0/ 0^on an bn cn
is smaller than / dx U . 0^ 0^ on an bn by a factor A. In this way we are able to establish a series approximation to I^. Now the proof of the Theorem rests on a
Lemma: If f is a slowly varying real function of n, thenn
/ / ^ ®an ®bn ® k n “bKn n n,K^O Proof: n p,n,n, K,K'
The property (E) of as given in Appendix 7.1 is used to evaluate the right-hand expression which gives
R.H.S. = S Æ + (2) + (D + (J)+©j, where
^ p=n=n ^ t ® k n ®bKn * W ®aKn ®bK+ 2w n * s' ®aKn ®bK- 2ir n^ '— —
®
■ „ î n w'4% : %
144.
2 . / . I- ' kn » M - 2. „') •
Since f^ is slowly-varying with p we may make the assumption
s £ l:i-~.jj.f
% £ s IzizM? = ^ £ ;
p^n ^ (p-n)^ ” pi^n (p-n)’ ”S £ . £ Z M f ' " = 0 . p4n P P-" " p4n P-"
Hence the f^/ in @ may be replaced by and
^ ~ ^ ^nl-4 ® k n ®bKn " 8 ®aKn ®bK+ 2u n ' 8 ®aKn ®bK- ^ '
L L
Similarly
^ ^ Cl-C-1) )(9aKn ®bKn^' 2 ®aKn ®bK+ Znr/ " T ®aKn^
*bK- 2ir
145
^ "n A "on ®a®b “ ^n ® k n ®bKn
n ivii
Soiïiè remarks on the above approximation are worth mentioning. Let y(x) be a function of x, then we can readily show
n-n/ (A) L < , n (B) L= < • 6 ■ . / y w u ^ d x n --- Æ
Therefore our approximation used in proving the Lemma is equivalent to the approximation
< l ^ F ( x ) - 0 and < ^ F(x) - 0 .
o X
where , .
F(x) = S ) .
Now to apply the Lemma to prove the Theorem we have
/L S £ y d x n U on an bn ej = /L 2 £ n
(/u
on a b 9 6, dx - Æ <0 > »a nn n
<®b>n) ^^0 ^n ® k n ®bKn ' ^he Lemma, n
Applying this result to
00
I = / dx -00 a = S I ,n n
-146-
7.8 Appendix 7« 3 Evaluation of the Hamiltonian
H = / d ^ x K., where Jt= Q^/(l+(Vçf K 1 + — G" Firstly approximation may be made for the derivatives of
/i 1 / . r -m " Km "'"59 " A m " L(4‘ -m> ) = 1 ^Kml Km — — - . 4 / -m‘ '591 = « ‘<lKm ^ L '^K^'m^m^ .
Since is assumed to be slowly-varying with m^ , the second sum in the above expression is very small and may be neglected in order to be consistent with the approximations made in Appendix 1 ,2
Now apply the Theorem to evaluate H„
n — Q — n a b n K^O ~ —
(7.8-1)
w h ere 9^ = (€ Ç 2" C g , c tt/Q )
may be expressed in terms of the metric tensors g_^ and g^*^ a b
i!â = l i l Ç H r f = ç l £ l - a'
Oj
00 ^
147-
% ■ Æ
'P'-
1+(VS): 00 g H - s QL^ ,/l+<?S>:,) ( 1 + ^ <T,>^) + i 2 (ql iK^ p^„ - v L n n" " _2KM
W
n' ^ ^ 00««»
l i s <rm ^Kn ^Kn uo «mftst.ia
^Kn'<*y.
^ '■" oj ok *r
^ " c 4 » i ’e - K^O j ,k . TK r TK, where g^ = [g = <^> — n148-
;
7.9. Appendix .7.4 The Approximately.Classical Nature of
Two assumptions are made in relation to the everywhere-slowly- varying nature of the field.
(1) an = 8^*^ - <6 > << 8^*^ and a a n a an << 1 in a neighbourhood of^ X = nL of dimensions of the order L. Note that 6 stand for thea dimensionless quantities Ç y — so that the above inequalities are independent of the units employed.
lie
(2) L can be large, or more precisely, we require that — « 1, where QL"
he— is a dimensionless quantity. QL"
- 7 - 3
In C.G.S, units with Q = 10” ei?g cm” « the mean energy density of the universe, we get L >> ^ 10 ^'^cm.
Firstly these assumptions enable us to count the orders of smallness of a quantity (see Appendix 7.2.) Secondly they lead to the result
T K ‘
that g^ may be regarded as c-numbers. This result is seen in the following analysis based on a finite difference method.
^*'1^ -^CïCx'+L, , x^) - S(x'-L, x“ , x^)) .
<S,l>n " r s 7 I ^ % n ^ ' Son-)' n+ = (n'+l, n % n^ ) ; n- =
” 2 L - -
(n‘ -1, n^ , n’ ).
149-