A Sensitive Test of Mass-Energy Relation

0.11 MB·English

by  Ying-Qiu Gu

Canonical Form of Field Equations
Ying-Qiu Gu
∗
Department of Mathematics, Fudan University, Shanghai, 200433,China
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Abstract
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In this paper, we derive a canonical representation for the first order hy-

7 perbolic equation systems with their coefficient matrices satisfying the Clifford
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algebra Cl(1,3), and then demonstrate some of its applications. This canonical

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formalism can naturally give a unified description for the fundamental fields in
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physics.
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PACS numbers: 11.10.-z, 11.10.Cd, 12.10.-g
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e Keywords:Clifford algebra, quaternion, canonical field equa-
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1 Introduction
Dirac derived his celebrated equation via the Clifford algebra Cl(1,3), but the re-
searches on this algebra in physical aspect seems to be limited at this simplest equa-
tionandtheabundant connotationsofCl(1,3)areoverlooked. Inmathematicalaspect,
the researches on the Clifford algebra are quite thorough and complete. The complex
and real irreducible representations of this algebra have been studied in details, and
complete results for the classification and the reduction are provided [1-9]. However
these studies are all in abstract level, and the results are inconvenient for physical
applications.
In this paper we give a concrete representation for Cl(1,3), then show an equiva-
lent canonical form of the first order hyperbolic equation with the coefficient matrices
∗
email: [email protected]
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satisfying Cl(1,3), and then display some of its applications. An early version of this
work once published in [10], but it is not easy to accessible. Here we give a review and
some improvements for this work.
According to some simple postulates, we find that the field variables u(xµ) =
(u ,u , ,u )T of a physical system satisfies the first order hyperbolic equation as
1 2 n
···
follows[11]
∂ u = A ∂ u+B(u,x,t), (1.1)
0 k k
where ∂ = ∂ , A ∂ = A ∂ + A ∂ + A ∂ , the index T stands for transposed
0 c∂t k k 1∂x1 2∂x2 3∂x3
matrix. Considering covariant principle, for any physical system, (1.1) should satisfies
the Lorentz invariance. This is in some sense equivalent to that the second order form
of (1.1) satisfies wave equation (∂2 ∂ ∂ )u = F(∂ u,u,x,t). For this reason, the
0 k k µ
−
following restriction condition for coefficient matrices is natural,
A A +A A = 2δ . (1.2)
k l l k kl
That is to say, we accept that the coefficient matrices satisfy the Clifford algebra
Cl(3,0). Under this condition, we will prove that u(xµ) must have even elements, and
there exists a linear transformation v = Q−1u, such that (1.1) can be transformed into
the following canonical form
∂ v = αk∂ v +b(u,x,t), (1.3)
0 k
where
αk = diag[σk,σk, ,σk; σk, σk, , σk], (1.4)
··· − − ··· −
0 1 0 i 1 0
σ1 =  , σ2 =  − , σ3 =  . (1.5)
1 0 i 0 0 1
     
−
Further more, for 3 +1 dimensional form Aµ∂ u = F(u), under the restriction of
µ
Clifford algebra Cl(1,3)
AµAν +AνAµ = 2ηµν, (1.6)
where we choice the Minkowski metric ηµν = diag[1, 1, 1, 1], then we have the
− − −
canonical form of dynamic equation
γµ∂ v = f(v), (1.7)
µ
where
0 αµ
γµ =  , (1.8)
αµ 0
 
e
and
α0 = I = α0, αk = diag[σk,σk, ,σk] = αk. (1.9)
··· −
e e
2
2 Proof of the Main Result
In this section, the tensor formalism and the Einstein’s summation are not adopted,
and all variables and parameters are regarded as local quantities for temporary use.
Lemma 1. Assume ρ = diag[I , I ], and I denotes n n identity matrix. Let
3 n n n
− ×
γ be a 2n 2n matrix, then
×
ρ γ γρ = 0 γ = diag[A ,A ], (2.1)
3 3 1 2
− ⇔
0 B
 1 
ρ γ +γρ = 0 γ = . (2.2)
3 3
⇔ B 0
 2 
where A ,A ,B ,B are n n matrices, stands for equivalence in logic.
1 2 1 2
× ⇔
Lemma 2. Assume q2 = I, i.e. q is an involutory matrix, then
0 A
 
qρ +ρ q = 0 q = , (2.3)
3 3
⇔ A−1 0
 
where A is an n n nonsingular matrix.
×
Proof. By (2.2), we have
0 A
 
q = . (2.4)
B 0
 
In addition by q2 = I, we get AB = BA = I . This finishes the proof.
n
Lemma 3. Assume the following equalities hold,
ρ2 = I, ρ ρ +ρ ρ = 0, (2.5)
1 1 3 3 1
then there exists a nonsingular matrix Q , such that
1
0 I
Q−1ρ Q = ρ , Q−1ρ Q =  n , (2.6)
1 3 1 3 1 1 1
I 0
 n 
where ρ = diag[I , I ]
3 n n
−
Proof: By (2.5) and Lemma2, we have
0 A
 
ρ = ,
1
A−1 0
 
where A is an n n nonsingular matrix. Let
×
Q = diag[I ,A−1], Q−1 = diag[I ,A], (2.7)
1 n 1 n
we get (2.6).
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Lemma 4. Assume
0 I I 0
 n   n 
ρ , ρ , (2.8)
1 3
≡ I 0 ≡ 0 I
 n   n 
then for nonsingular Q we can check
2
Q ρ = ρ Q and Q ρ = ρ Q Q = diag[C,C]. (2.9)
2 1 1 2 2 3 3 2 2
⇔
Lemma 5. Assume ρ2 = I, then we have
2
0 E
 
ρ ρ +ρ ρ = 0,( k = 1,3) ρ = i − , (2.10)
2 k k 2 2
⇔ E 0
 
where ρ and ρ are defined by (2.8), E2 = I is an involutory matrix.
1 3 n
Proof: It suffices to prove the part “only if ”. Noticing ρ ρ +ρ ρ = 0 and (2.2),
2 3 3 2
we have
0 B
 1 
ρ = . (2.11)
2
B 0
 2 
In addition by ρ ρ +ρ ρ = 0, we get B = B . Let B = iE, then the result holds
2 1 1 2 1 2 2
−
by ρ2 = I.
2
Lemma 6. Let ρ and ρ be defined by (2.1) and ρ by (2.10), then there exists a
1 3 2
nonsingular matrix Q , such that
3
Q−1ρ Q = ρ ,( k = 1,3), (2.12)
3 k 3 k
0 D
Q−1ρ Q = i − , (2.13)
3 2 3
D 0
 
where D = diag[Im, In−m].
−
Proof: Let Q = diag[K,K], where K is a nonsingular matrix, then (2.12) holds
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by lemma 4. Let E be given by (2.10), and assume that E have m eigenvalues equal
to 1 , and n m eigenvalues equal to 1. So we can find a nonsingular matrix K such
− −
that
K−1EK = diag[Im, In−m] D. (2.14)
− ≡
Moreover
−1
K 0 0 E K 0 0 D
Q−1ρ Q = i   −   = i − . (2.15)
3 2 3
0 K E 0 0 K D 0
      
This finished the proof.
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Theorem 1. Suppose that matrices A satisfy
k
A A +A A = 2δ I, ( k,l = 1,2,3), (2.16)
k l l k kl
∀
then there exists a nonsingular matrix Q such that
Q−1A Q = diag[ξk, ζk], (k = 1,2,3), (2.17)
k
−
where
m
ξk = diag[σk,σk, ,σk], (2.18)
z }| {
···
n−m
ζk = diag[σk,σk, ,σk], (2.19)
z }| {
···
and m(0 m n) is independent of k.
≤ ≤
Proof: We prove the theorem by five steps.
1◦. By (2.16) we have A2 = I, so we can find a nonsingular matrix P such that
k 1
P−1A P = diag[I , I ] = ρ , (2.20)
1 3 1 n n 3
−
or
P−1A P = diag[I , I ,I], (2.21)
1 3 1 N N
−
or
P−1A P = diag[I , I , I]. (2.22)
1 3 1 N N
− −
In what follows, we will prove that (2.21) or (2.22) contradicts the condition (2.16).
We prove the results by contradiction. Assume (2.21) holds, Let
F F
P−1A P =  1 2 . (2.23)
1 1 1
F F
 3 4 
where F is a 2N 2N matrix. Substituting (2.21) and (2.23) into A A +A A = 0,
1 1 3 3 1
×
we get
0 f 0
  1   
F = , F = ,
 1  f1−1 0  2  f2  (2.24)
F = (0,f ), F = 0,
 3 3 4
where f is a N row matrix, f is a N column matrix. In addition by A2 = I and
2 3 1
(2.24), we have
F2 +F F = I, F F = I, (2.25)
1 2 3 3 2
F F = 0, F F = 0. (2.26)
1 2 3 1
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By (2.26) we have F = F = 0, this is a contradiction to the equality F F = I, so
2 3 3 2
(2.21) doesn’t hold. Similarly, (2.22) also doesn’t hold.
2◦. By Lemma 3, we can find a nonsingular matrix P such that P−1ρ P = ρ , and
2 2 3 2 3
0 I
P−1(P−1ρ P )P =  n  = ρ
2 1 1 1 2 1
I 0
 n 
3◦. By Lemma 6, we can find a nonsingular matrix P such that
3
P−1ρ P = ρ , (k = 1,3),
3 k 3 k
0 D
(P P P )−1A (P P P ) = i −  = ρ ,
1 2 3 2 1 2 3 2
D 0
 
where D = diag[Im, In−m].
−
◦
4 . Define permutation matrix
0 0 1
   
jk diagIj−1, 0 Ik−j−1 0 ,I2n−k, (2.27)
P ≡    
  1 0 0  
   
l
 2k,n+2k−1, if n = 2l,
P4−1 =  kQ=l1P 2n−1 (2.28)
2k,n+2k−1 j,j+1, if n = 2l+1,
 kQ=1P jQ=n P
then we have
P−1ρ P = diag[σk,σk, ,σk], (k = 1,3), (2.29)
4 k 4
···
m n−m
P−1ρ P = diag[σ2,σ2, ,σ2, σ2, σ2, , σ2]. (2.30)
4 2 4 z }| { z }| {
··· − − ··· −
5◦. Let
Q = P P P P diag[I ;σ2,σ2, ,σ2], (2.31)
1 2 3 4 2m
···
then (2.17) holds.
As corollaries of Theorem 1, we have equation (1.3) under condition (1.2), and
equation (1.7) under condition (1.6).
3 Application
In what follows, we expand some elemental cases of (1.7) with linear f(v). The results
are well known, but the viewpoint seems to be enlightening.
For (1.7), we define quaternionic partial operators
∂ = αµ∂ , ∂ = αµ∂ . (3.1)
µ µ
e
e
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Considering the covariant of the equation (1.7), we find for any linear case, (1.7) must
take the form similar to [10]
∂Ψ = k Ψ,
 1
(3.2)
 e
∂Ψ = k Ψ,
2
 ee
where Ψ and Ψ have the same number of elements but obey different transformation
e
rule. That is to say, under the restriction of Clifford algebra Cl(1,3), the physical fields
always appear in couples.
A. Dirac Equation(fields with spin 1)
2
The least system of (3.2) is the case αµ = σµ. For this case, if k k = 0, making
1 2
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transformation
ψ = k Ψ, ψ = k Ψ, m = i k k ,
q 2 q 1 q 1 2
e e
we get Dirac equation
i∂ψ = mψ,

(3.3)
 e
i∂ψ = mψ.
 ee
B. Maxwell equation(fields with spin 1)
The case of (3.2) just more complicated than (3.3) is that Ψ and Ψ have 4 elements,
e
or the coefficient matrix (1.9) consists of two Pauli matrices, i.e.,
a a b b
 1 3   1 3 
A = = Ψ, A = = k Ψ, (3.4)
1
a a − b b
 2 4  e  2 4  e
then the field equation (3.2) with source eQ is equivalent to
∂A = A,

− (3.5)
 ∂A = m2eA eQ,
−
 ee
where e denotes the coupling coefficient(charge).
For the case m = 0, expressing A, A and Q by quaternion, i.e.,
e
A = Aµσ , A = Hµσ , Q = qµσ , (3.6)
µ µ µ
e
where σ = (σ0, σ1, σ2, σ3) is taken as quaternionic basis, then (3.5) is equivalent
µ
− − −
to the following vector formalism
βµ∂ V = H,
 µ
− (3.7)

βµ∂ H = eq,
µ
−
 e
where β0 = I, and
0 1 0 0 0 0 1 0 0 0 0 1
     
1 0 0 0 0 0 0 i 0 0 i 0
β1 =  , β2 =  − , β3 =  , (3.8)
 0 0 0 i   1 0 0 0   0 i 0 0 
     
     − 
 0 0 i 0   0 i 0 0   1 0 0 0 
     
−
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V = (A0,A1,A2,A3)T,

 Hq ==(q(H0,q0,1H,q12,,Hq32),TH, 3)T,
 βeµ = (β0,−β1,−β2,−β3).
If the following conservation law holds, which is inevitable in a larger coupling
system
∂ qµ = ∂ Aµ = 0,
µ µ
then the 3 d vector form of (3.7) becomes standard Maxwell equation[12]
−
∂ qµ = ∂ Aµ = 0, ∂α∂ Aµ = eqµ,
 µ µ α
 −→E =−→E−=∇eAq00,−∂0−→A, −→B =−→E∇=×−→A∂,0−→B, (3.9)
∇· ∇× −
 ∇·−→B = 0, ∇×−→B = ∂0−→E +e−→q ,
where
−→H −→E +−→Bi, −→A (A1,A2,A3), q (q1,q2,q3). (3.10)
−→
≡ ≡ ≡
The formalism −→H = −→E + −→Bi is once noticed by several authors, but there it was
artificially constructed[13, 14, 15, 16, 17]. By (3.7) we learn that the magnetic charge
should be the imaginary part of the complex coefficient e. But in the coupling system
of spinor and vector, the stability of the system requires the charge e in (3.5) or (3.7)
to be a real. This requirement is equivalent to the assertion that the magnetic charge
should be absent in the Nature. For the case m = 0, (3.5) seems to be meaningful in
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describing the strong interaction.
In curved spacetime, denote the generalized form of βa and βa by
e
gµν = hµ hν ηab, ρµ = hµ βa, ρµ = hµ βa.
a b a a
e
e
Then the Maxwell equation (3.7) have the following Dirac-like formalism
(ρµ∂ +M)V = H,
 µ
− (3.11)

(ρµ∂ +N)H = eq,
µ
−

e
where V, H are vectors projected to the tetrad frame,
M = 1 ρµJνρ ρ + 1ρ ρ Jνρµ 1ρ Jµ,
 12 ν µ 6 µ ν − 2 µ (3.12)
 N = 1 ρeρ Jνρµ + 1ρeνJeµρ ρ 1Jeµρ ,
12 µ ν 6 µ ν − 2 µ

e e e
and matrices Jµ take the following form
1
J = K I + ∂ g (ραρβ ρβρα), (3.13)
µ µ α µβ
4 −
e e
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here K is a number similar to ∂ √g.
µ µ
C. Tensor equation(fields with spin 2)
Similar to (3.7), we have the linear dynamic equation of tensor as follows
βµ∂ G = F,
 µ
(3.14)

βµ∂ F = λG+κT,
µ
 e
where G = (Gµν) is a covariant tensor in matrix form, T = (Tµν) is the source tensor,
F = G is the complex field intensity. Whether the generalized form of (3.14) in curved
e
spacetime is equivalent to Einstein equation is unknown, because the gravitation is
quite different from the matter fields. For the matter fields u, the affine connection is
independent of ∂ u. But for the metric tensor (gµν), we can not realize this intention
µ
via (3.14). However the Einstein’s equation can be expressed by a Clifford algebra
isomorphic to a direct product of two quaternion algebras[18]. The quaternion algebra
should play a more important role in physics.
An integrative application of system (1.7) or (3.2) is given in [19].
Acknowledgments
The author wishes to thank Academician Ta-tsien Li, Prof. Guang-Jiong Ni and Prof.
Han-Ji Shang for guidance, thank Prof. Hao Wang, Prof. Er-Qiang Chen and Mr.
Hai-Bin Lei for their encouragement and help.
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