Local Generalized Pauli's Theorem

0.1 MB·English

by  Nikoly Marchuk

Local Generalized Pauli’s Theorem
Nikoly Marchuk and Dmitry Shirokov
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0 January 25, 2012
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Abstract
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In this work a generalized Pauli’s theorem (proved by D. S. Shi-
]
h rokov for two sets ea,ha, a = 1,...,n of Clifford algebra elements
p
- Cℓ(p,q), p + q = n) is extended to the case, when one or both sets of
h
t elements depends smoothly on points x of the Euclidian space. We
a
m prove that in this case for any point x0 of Euclidian space there exists
[ ε-neighborhood Oε(x0) and there exists smooth function T : Oε(x0) →

1 Cℓ(p,q) such that two sets of Clifford algebra elements are connected
v by a similarity transformation ha(x) = T(x)−1ea(x)T(x), a = 1,...,n
5
8 in Oε(x0).
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1. Let V be r-dimensional Euclidian space with scalar product (x,y), ∀x,y ∈
0
V and with the norm
2
1
:
v kxk = (x,x), ∀x ∈ V.
p
i
X
r
Let Ω be domain in V and let ε > 0 be positive real number. ε-
a
neighborhood of a point x ∈ V is the domain
0
O (x ) = {x ∈ V : kx − x k < ε}.
ε 0 0
F R F C F
Consider a real or complex ( = or = ) Clifford algebra Cℓ (p,q) of
1 a n
dimension n = p + q with generators e , a = 1,...,n and with basis of 2
elements
a ab 1...n
e,e ,e ,...,e , (1)
1An important special case is r = n, when the basis of V is the set of generators ea.
In this case V can be considerd as a pseudo-Euclidian space with two metrics - Euclidian
and pseudo-Euclidian. We use Euclidian metric to determine a neighborhood of a point.
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enumerated by ordered multi-indices of length from 0 to n.
The generators satisfy the relations
a b b a ab
e e + e e = 2η e,
ab
where η are elements of the diagonal matrix η of order n with p elements
equals 1 and q elements equals −1 on the diagonal.
Consider a function
F
f : Ω → Cℓ (p,q)
F
with values in Clifford algebra Cℓ (p,q). The function f = f(x) can be
written in the form
a 1...n
f = ue + u e + ... + u e ,
a 1...n
F
where u = u(x), u = u (x), ... are functions Ω → and basis elements (1)
a a
do not depend on x ∈ V .
If functions u,u ,...,u have continuous derivatives up to order k in
a 1...n
Ω, then we say that (real or complex) functions u,u ,...,u and f belong
a 1...n
k 0
to the class C (Ω) (C (Ω) – the class of continuous functions in domain Ω).
Theorem 1
(Local generalized Pauli’s theorem in the case of even n).
a a
Let n be even positive number and h = h (x), a = 1,...,n are functions
F k
Ω → Cℓ (p,q) of class C (Ω) such that
a b b a ab
h (x)h (x) + h (x)h (x) = 2η e, a,b = 1,...,n, ∀x ∈ Ω.
Then for any x ∈ Ω there exists ε > 0 and there exists a function
0
F
T = T(x) : O (x ) → Cℓ (p,q), satisfying the conditions
ε 0
k
1. T(x) – function of class C (O (x ));
ε 0
F
2. T(x) – an invertible element of Clifford algebra Cℓ (p,q) for any x ∈
O (x );
ε 0
3. ea = T−1(x)ha(x)T(x), a = 1,...,n, ∀x ∈ O (x );
ε 0
4. The function T(x) is defined up to multiplication by (real in the case
F R F C k
= or complex in the case = ) function of class C (O (x )) that
ε 0
is not equal to zero for any point of O (x ).
ε 0
2
Proof A
. We denote elements of the basis (1) by e , where A are ordered
multi-indices of length from 0 to n. Let denote e = (eA)−1. For each basis
A
A A 2 A 2 A
element e we have (e ) = e or (e ) = −e, so for all A we have e = ±e .
A
We denote
ab a b
h = h h , for 1 ≤ a < b ≤ n;
abc a b c
h = h h h , for 1 ≤ a < b < c ≤ n
and so on. Let denote elements
a ab 1...n
e,h ,h ,...,h
A
by h .
Let consider sum of the following form (sum over all ordered multi-indices
A of length from 0 to n)
A
Xh (x)FeA (2)
A
where F is an element of the basis (1).
Consider an arbitrary point x ∈ Ω. By generalized Pauli’s theorem [1]
0
we have at least one basis element (1) (denote it by F) for which
A
T = Xh (x0)FeA 6= 0. (3)
A
We define the norm of Clifford algebra elements by
|U| = Tr(U†U),
p
where the operation of Hermitian conjugation † is defined in [2]. Then from
(3) we obtain
A
|T| = |Xh (x0)FeA| = δ > 0.
A
k
Since a linear combination of functions of class C (Ω) is a function of class
k A
C (Ω), then the expression | h (x)Fe | is a continuous function of x ∈ Ω.
PA A
Then there exists a real number ε > 0 such that
A
|Xh (x)FeA| > δ/2, ∀x ∈ Oε(x0)
A
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in ε-neighborhood of the point x .
0
Consequently, we construct a function
A
T(x) = Xh (x)FeA 6= 0, ∀x ∈ Oε(x0)
A
k
of class C (O (x )). By generalized Pauli’s theorem [1] we have
ε 0
ea = T−1(x)ha(x)T(x), a = 1,...,n, ∀x ∈ O (x ))
ε 0
or, equivalently,
ha(x) = T(x)eaT−1(x), a = 1,...,n, ∀x ∈ O (x )).
ε 0
The theorem is proved.
The theorem evidently generalizes to the case when both sets of Clifford
algebra elements depend on points x of the Euclidian space.
Let formulate and prove a similar theorem for the case of odd n.
Theorem 2
(Local generalized Pauli’s theorem in the case of odd n for real
Clifford algebra).
a a
Let n be positive odd number and h = h (x), a = 1,...,n are functions
R k
Ω → Cℓ (p,q) of class C (Ω) such that
a b b a ab
h (x)h (x) + h (x)h (x) = 2η e, a,b = 1,...,n, ∀x ∈ Ω.
1 2 n 1...n
Then the product h (x)h (x)...h (x) = h does not depend on x and
1...n
equals ±e or ±e (last case is possible only if p − q = 1 mod 4).
Then for any point x ∈ Ω there exists ε > 0 and there exists a function
0
R
T = T(x) : O (x ) → Cℓ (p,q), satisfying the conditions
ε 0
k
1. T(x) – function of class C (O (x ));
ε 0
R
2. T(x) – an invertible element of Clifford algebra Cℓ (p,q) for any x ∈
O (x );
ε 0
3. (a) ea = T−1(x)ha(x)T(x) ⇔ h1...n = e1...n,
(b) ea = −T−1(x)ha(x)T(x) ⇔ h1...n = −e1...n,
(c) ea = e1...nT−1(x)ha(x)T(x) ⇔ h1...n = e,
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(d) ea = −e1...nT−1(x)ha(x)T(x) ⇔ h1...n = −e,
where equalities hold for a = 1,...,n and ∀x ∈ O (x );
ε 0
4. The function T(x) is defined up to multiplication by elements λ(x)e +
1...n k
ν(x)e , where λ(x) and ν(x) are real functions of class C (O (x ))
ε 0
1...n
such that element λ(x)e+ν(x)e is invertible for any point of domain
O (x ).
ε 0
Before discussing the proof of this theorem let us formulate another the-
orem.
Theorem 3
(Local generalized Pauli’s theorem in the case of odd n for
complex Clifford algebra).
a a
Let n be positive odd number and h = h (x), a = 1,...,n are functions
C k
Ω → Cℓ (p,q) of class C (Ω) such that
a b b a ab
h (x)h (x) + h (x)h (x) = 2η e, a,b = 1,...,n, ∀x ∈ Ω.
1 2 n 1...n
Then the product h (x)h (x)...h (x) = h does not depend on x and
equals ±e (in the case p − q = 1 mod 4) or ±ie (in the case p − q = 3
1...n
mod 4) or ±e (in both cases).
Then for any point x ∈ Ω there exists ε > 0 and there exists a function
0
C
T = T(x) : O (x ) → Cℓ (p,q) such that
ε 0
k
1. T(x) – a function of class C (O (x ));
ε 0
C
2. T(x) – an invertible element of Clifford algebra Cℓ (p,q) for any x ∈
O (x );
ε 0
3. (a) ea = T−1(x)ha(x)T(x) ⇔ h1...n = e1...n,
(b) ea = −T−1(x)ha(x)T(x) ⇔ h1...n = −e1...n,
(c) ea = e1...nT−1(x)ha(x)T(x) ⇔ h1...n = e,
(d) ea = −e1...nT−1(x)ha(x)T(x) ⇔ h1...n = −e,
(e) ea = ie1...nT−1(x)ha(x)T(x) ⇔ h1...n = ie,
(f) ea = −ie1...nT−1(x)ha(x)T(x) ⇔ h1...n = −ie,
where equalities hold for a = 1,...,n and ∀x ∈ O (x );
ε 0
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4. The function T(x) is defined up to multiplication by elements λ(x)e +
1...n k
ν(x)e , where λ(x) and ν(x) - are complex functions of class C (O (x ))
ε 0
1...n
such that element λ(x)e+ν(x)e is invertible for any point of domain
O (x ).
ε 0
Proof
. Proofs of Theorems 2 and 3 are similar to the proof of Theorem
1. We must use generalized Pauli’s theorems [1] for Clifford algebras of odd
dimension n. We consider expressions (instead of the expressions (2))
A
X h (x)FγA,
A∈IEven
where sum is over multi-indices of even length I = {A, |A| − even}.
Even
B
Element F is always among the basis elements {e } or among the ex-
B C
pressions {e +e } of sum of two basis elemebts. So, F does not depend on
the point x.
All other considerations are similar to considerations for the case of even
n. The theorems are proved.
Theorems 2 and 3 evidently can be generalize to the case when both sets
of Clifford algebra elements depend on points x of the Euclidian space.
References
[1] Shirokov D.S., Extension of Pauli’s theorem to Clifford algebras, Dokl.
Math., 84, 2, 699-701 (2011).
[2] Marchuk N.G., Shirokov D.S., Unitary spaces on Clifford algebras, Ad-
vances in Applied Clifford Algebras, Volume 18, Number 2, pp.237-254
(2008).
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