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1
УДК 539.192 (043.3)
Klimko G.T.
Donetsk National Technical University (DonNTU), Ukraine
Ukraine, 83001, c. Donetsk, Artema st., 58, [email protected]
Recovery of the basis of group π2n representation on its
subgroup πn×n and Harriman’s theorem
Климко Г.Т.
Донецкий национальный технический университет, Украина
Украина, 83001, г. Донецк, ул. Артема, 58, [email protected]
Восстановление базиса представления группы π2n на
её подгруппе πn×n и теорема Гарримана
Климко Г.Т.
Донецький національний технічний університет, Україна
Україна, 83001, м. Донецьк, вул. Артема, 58, [email protected]
Відновлення базису, що дає представлення групи π2n
на базисі її підгрупи πn×n , і теорема Гаррімана
The proof of Garriman’s theorem [1] is given for arbitrary order reduced density matrix of
both the clear, and the mixed states of fermions at once. Essential its parts are a Pauli's exclusion principle, rotation group symmetry of spin functions and new commutation relations.
Теорема Гарримана [1] доказана для редуцированных матриц плотности чистого и
смешанного состояний фермионов из принципа Паули, симметрии спиновых функций,
новых следствий из перестановок штрихованных и не штрихованных переменных.
Теорема Гаррімана [1] доведена для редуцироних матриць густини чистого і змішаного
станів ферміонів з використанням принципу Паулі, симетрії спінових функцій і нових
наслідків, пов'язаних з переміщеннями штрихованих і нештрихованих змінних
1 Reduced density matrix
Reduced density matrix of arbitrary order p (RDM-p) [2, 3], including transitional
RDM-p, for fermions states with spin projection values M and M can be written in form [4]
Γ ( p) 
 x , x ,..., x
1
2
p
x1, x2 ,..., xp
  () R((p))μ Tˆ((p))μ .
Its spin and “conjugated” space components look like
(1)
2
 ( p) μ
T(γ)ω   
σ pit

t  i t  i t  
pit  μ σ 
(2)
 pit it  μ pγ ωμ ,
   
(p)μ
R
      1ε i t  ε i t  μ p γ ωμ pit it  μ 
γ  t i i
t tμ

 p  
 Lμt
i t * jp  t it  μ* jp  t  μ
There are conjugated unitary transformation coefficients
p γ ωμ pit it  μ

.
p
i t it  μ pγ ωμ
(3)
and
are appeared in Eqs (2), (3). The contravariant spinors
σ p it  α(i1 )  α(i2 )  α(it )  β(j1` )  β(j2 )  β(j p t ) ,
its conjugated covariant spinors and hermit spin tensors
Tˆ((p)μ
γ)ω
(4)
of a rank ω are ortho-
normalized. Antisymmetry of wave functions of states is concluded in Eq. (1). Therefore for
2
antisimmetrizer Aˆ  Aˆ is true
 Rp  Tˆ p  Aˆ   Rp  Tˆ p  Aˆ . Both spin
 
 
and space coordinates of particles are designated by its numbers it   j p t  1, 2,... p .
2. Recovery of the basis of group π2n representation on its
subgroup πn×n
Unlike the one-particle contravariant spinors α and β, which connected to spin space irreducible representation (IR) D ½ of rotation group SU  2 [5, 6], covariant spinors α+(i) and
β +(i) belong to the equivalent IR D½* = V ½∙D½∙(V ½) + with unitary matrix [5, page 165]: V ½

α 
0 1 
½ + 

= 
.
Then
the
spinors
(V
)
∙

 β  =
1
0


 
 β 
   are transforming on space of IR D ½, as
 α 


3
α
well as spin functions   . Therefore conjugated to Eq. (4) covariant spinors are equivalent
 β
to contravariant ones. After replacement every α + (i) on a spinor –β(i) and β + (j) respectively
on α(j) its generate the basis of spinors σ p i t . It’s belong the same space of a direct product

p
i 1
1
Di 2 of p identical IR D
1
2
of group SU  2 , as contravariant spinors (see Eq. (4)),
σ p i t   1  β(i1 )  β(i2 )  β(i pt )  α(j1 )  α(j2 )  α(j pt ) .
t
(5)
As result, the basis from products of kontra- and of covariant spinors, as the basis of
spinors of a 2p-rank from products of only contravariant spinors σ p it
σ  p it  , be-
long to the same representation of rotation group. It allows properties of one basis, established
on the ground of its group-theoretical reviewing, to transfer on properties of equivalent basis.
From theory of IR
functions
    p  , p    for permutation group  2 p
it follows, that the
u  Yˆu Qˆuv and function v  Yˆv , where permutation Qˆ vu  Qˆuv1 substitutes
numbers of coordinates in Young table of u on numbers from the Young table
similate one IR basis. Instead the permutation Q̂uv connects functions
v
[5, 6], as-
 u and  v [6]
Qˆuvv  QˆuvYˆv  QˆuvYˆvQˆuv1Qˆuv  YˆuQˆuv  u .
(6)
Call to mind, that both in lines and in columns of standard Young tables [5, 6] the number of coordinates are ranking from left to right and from top to down according to increase of
their place number in the initial list. For list 1, 2, …, p-, p-+1,…,p, 1, 2, p-, p-+1′,…,p′
such Young table will be obtained, when the coulombs of Young scheme [p+, p-], where
ω = 0, 1, …, р, are filling by numbers 1, 1, 2, 2,…, p-, p-, p-+1,…,p, p-+1′,…,p′
1
2
…
im
…
p-
1′
2′
…
i′m
…
(p-)′
p-+1
Fig. 1.
…
p
(p-+1)′
…
p′
4
The number of standard Young tables [5, 6] equals to dimension of IR  p   , p   
 2p   2p 
 2 p ! 2  1
f2 p  
,


 p     p    1  p    1! p    !
(7)
A permutation may be represented by product not intersected cyclical permutations. Its
do not leave invariant variable places from their list. Cycles are representing by product of
transposition, which are not commute. Such transpositions commutate according to a rule [6]
 i, k    k , l    i , l    i , k    k , l    i , l  ,
(8)
We prove now main assertion according to which it is possible to receive any function
 u simmetrized on the standard Young table u for IR  p   , p    from function  v
permuting its arguments only inside each of sets on p variable with the primed or not primed
numbers separately, if in the second line of the “standard” table of Young v the variables of
one type set only remain, as in the Young table v in a Fig. 1. Clearly, if transition from
 v to
 u demands rearrangement of primed variables with not primed, then in Eq. (6) Qˆ uv   2 p .
ˆ
Really, it is possible to separate exactly luv transpositions Q
l uv 
  p  1  i, i 
luv
i 1
ˆ   , conserving after their operation on the table v correct regularity of not primed
in Q
uv
2p
numbers concerning primed both in table lines and in table columns. The number pairs luv of
permuted primed and not primed variables is restricted an inequality luv  min  p   ,  p 2 ,
where  p 2 – the whole part from p/2.
The factorization of permutations Qˆ uv  Qˆ uv   Qˆ uv   Qˆ l uv reduces our assertion to the
proof of a possibility of replacement of each transposition
imik  in Qˆ  
operation on Yˆv of permutations belonging to only a subgroup
Yˆ by an identical
l uv v
 p   p :
 ik , im   Yˆv  Yˆik ,im v   ik , im   2 Aˆik im Aˆik im Yˆv .
In equality (9) we always have k < m and as in Eq. (6) Young projectors are equal
(9)
5
p 
Yˆv  c   Aˆi i Sˆ
i i  p
 1
Yˆik ,im ·v  c 
There
p

Sˆi
p
ˆ A
ˆ
A



1
 k ,m
i i
 Sˆ
 Aˆ  
ik im
ˆ
... p S... p
i i
p
,
(10)
ˆ Sˆ
ˆ
A
...p \ im ,i  S
...p \i  ,im 
i im 
k p
k
k
p
. (11)

is product of antisimmetrizers and of simmetrizers re-
spectively, acting on the corresponding sets of variables, c is constant to ensure that Yˆv 2 = Yˆv .
ˆ in (6) may be changed on equal operator but only from
Due equality (9) any operator Q
uv
permutations belonging to subgroup
 p   p . It constructs group  2 p
the subgroup operators. If ik , im  ... p  ,
ˆ Sˆ
=A

ik im
...
Sˆ...
p
p
k
p 
1
Aˆik im Aˆik im
2


1

ik , im  ... p  , then Aˆik im Sˆ... Sˆ...
p
p
 0 and in a right-hand part of Eq. (9) instead of Aˆi i  Aˆimim · Yˆv the
kk
   i , i    i
transpositions
basis {  u } using
k
m
, im   4
will stay, when the definition (10) for Yˆv is used:
Aˆi i     ik ik    imim    Sˆ...
p
Sˆ...
p
.
(12)
  k , m
Here ... p  and ... p 
are the same sets of variables, as in Eq. (10).
Take in mind the equality:
Sˆ...
p
Sˆ...
p
  imik   Sˆ
.... p \im ,ik  p ... p \ik ,im p  m k 
Sˆ
i i . ,
(13)
and rules (8) we transform expression (12) into an operator
Aˆ i k i m Aˆ i k i m 1
2
p-ω
 Aˆi i   ik im ik ik   ik im imim   Sˆ...
 1
 
ν  k , m 
 Sˆ... p  \ i k , i m 
p 
\ i m , i k p 

(14)
p 
imik  ,
In Eq. (14) first transpositions are retracted into suitable antisimmetrizers with a change of
sign, and second are retracted into simmetrizers. Transposition
imik  will be commutated to
6
left with all operators transforming its. As a result we obtain the left-hand part of identity (9).
For the function
 M   n  M ! n  M !


to give the IR n  s, n  s basis of group
u  
Here
1 2
 2n
  ... n M   ... nM [6, 7] we always can
in the following form
 Cuv Qˆ1Qˆ 2  Qˆ1Qˆ 2  Yˆv   M ,

ˆ
 s  M  s.
Q1ˆ n  M Qˆ 2ˆ n  M

(15)

Cuv Qˆ1Qˆ 2 are simple numerical coefficients and u is a standard Young table.
3 The proof of the Garriman’s theorem
For the given p and  = M – M= t – t ´ in Eq. (3) the number of "independent" space
components RDM is determined both impossibility of build-up of tensors of a higher rank,
than ω = p, in p–partial spin space, and Garriman’s theorem, proved for some cases [1]. It defines a possibility to receive all components R((p))μ in Eqs. (1), (2) from "standard" component
with the same  and , which obtained according to a "standard addition schema of the spin
moments (o)” [5, 6] , that is equivalent Young table v in a Fig. 1. The obtained above results
(see Eq. 15)) allow us to give the common proof of the Garriman’s theorem.
To build the irreducible tensors Eq. (2) routinely the products of the conjugated spinors
, ,  and  must be changed by one-particle operators [8],
Iˆ          , Sˆz  tˆ1(0)  12 (        ),
(16a)
S   2  t1( 1)      , S  2  t1( 1)     ,
(16b)
Spσ Iˆ2  2, Spσtˆ1(μ)  tˆ1(μ)  
 μμ
2
, Spσtˆ1μ  Iˆ  0.
(16c)
(μ )
Here Iˆ is a unit operator, tˆ1` are components of one-particle tensor of the first rank. Then
the complete set of addition schemes {()} may be realized by serial adding the operator Iˆ or
tˆ1`(μ ) to already built irreducible tensor of smaller number of particles. As result, we can define
the union list from p one-particle operators and from the intermediate values of tensor rank for
addition schema ().
7
The operators σ p it
σ p i t  , as in Eq. (5), and Tˆ γ ω
spinors σ p it
(p)μ
Tˆ(γ)ω
can be transformed on the
p it  σ and the tensor
(2p)μ
depending out of 2p of spin variables.
Then last save the same addition schema [] = (), the moment  and its projection . It is
true, “as from the permutation operators of spin coordinates of particles commute with the
spin rotation operators [6] it follows that the dimension Eq. (7) of
groups
2p
 p  , p   
IR for
and the number of addition schemes {[]} of ,  spins with same complete val-
ues  and  are equal.” Each such addition schema correspond the line of IR of permutation
group
 2 p . Therefore 2p-particle functions Tˆ γ(2p)μ
ω
one set {[]}. They form basis for IR
can be numbered by addition schemes of
 λω  =  p  , p    of symmetric group  2 p
and ac-
 2 p 
cording to Eq. (15) they can be obtained from a function Tˆ 0  . Let's find mutual conformi 2 p 
ˆ
ty between spinor T 0 
 p 
ˆ
and tensor T 0  . It is evident the tensor
T(1(p)μ
with a special
p ω
...)ω
addition schema (o) = (1p-…)ω gives this appropriation. It looks like
Tˆ (pp)ωω
(1

...)ω
2
p ω
2 Iˆ(1)Iˆ(2)  Iˆ(p  ω)Sˆ (p  ω 1)  Sˆ (p).


(17)
It has the predicted values of ranks of the all intermediate tensors for any p and after using
Eqs. (16), (5) it is equivalent to a function-spinor of a rank 

Tˆ (2p p)( )  2
[1
...]ω
p 
2
p 
  ( (i )  (i)   (i) (i))
i 1
p

 ( j ) ( j ) .
j  p  1
(18)
Function (18) is really simmetrized under the Young table [6], which columns are completed
by numbers 1, 1, 2, 2, …, p-, p-, …, as in a Fig. 1, and from Eq. (15) follows
T[(2]ωp) 
p
C 

Qˆ ,Qˆ  p  p
ˆ ˆ T (2 p)  ˆ  p T (2 p) .
 Q,QQQ
  [1 p  ..]ω
[1 p  ...]ω
(19)
After we return to tensors Eq. (2), using Eqs. (16) in Eqs. (18), (19), the same expan(p)
sion remains valid both for the highest component T̂(γ)ω and for all components of the tensor
8
(p)μ
(p)
Tˆ(γ)ω
too, as they may be obtained from T̂(γ)ω by spin projection decreasing operators Ŝ 
 Sˆ , Tˆ p       1      1  Tˆ p  1 .


(20)
p
As operators Sˆ   Sˆ  i  commute with all permutation operators of particles and with
i 1
Qˆ  Qˆ    p   p too, we have general proof of “Garriman’s theorem” in spin space
Tˆ   
( p )( )
ω
 p
C 

Q,Q p  p
ˆˆ
 Q,QQT

( p )(  )
ˆ   ˆ  Tˆ
Q
 

p
1 p  ... ω

( p )(  )

1 p  ... ω
(21)
.
These tensors may be orthonormalized, as in (1), without change its obtained structure (21).
In this case applying expansion (2), we make up the identity (21) into linear relationships between coefficients p it it  μ p  γ  ωμ inside of Eqs. (2), (3). They contain same
factors, as in Eq. (21),
p it it  μ p  γ  ωμ 

p
C   Q, Q  pQˆ 1it Qˆ 1it  μ p 1p-ω ... ωμ .

Q, Q   p p
(22)
Taken into account that the number sets in Eq. (22) and ones at sign factor in Eq. (3) are
same we shall receive the connection of space components
RDM-p
R(pp)μ
R(p)(ω ) 
(1
...)
R((p))μ
with standard component
, which is similar to Eq. (21):
 1

Qˆ , Qˆ  ˆ p  ˆ p
q  q
p
ˆ ( p )(  ) Qˆ   ˆ  p   Rˆ ( p )(  )
C   Q, Q  QR
p 
p 
 
1 ...ω
1 ...ω .(23)
,
 p
where ˆ   p   p , (-1)q+q' is sign factor for permutation Q·Q'. Moreover, it is true that




R(p)μ
 Sp Γ  p  x1 ,..., x p x1,..., xp  Tˆ p  ,
γ 
(24)
if spin tensors (21) are orthonormalized. This simple process always can be applied to obtain
exact result type Eq. (23) for any RDM. It completes generally proof of Garriman’s theorem.
9
References
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