Generalized zilch and the jordan canonical form

LINEAR

ALGEBRA

Generalized

AND

ITS

APPLICATIONS

19

(1973)

6, 19-36

Zilch and the Jordan Canonical Form

J. HEADING University

College of Wales

Aberystwyth,

U.K.

Communicated

by J. H. Wilkinson

ABSTRACT The concept

of zilch, previously

certain quadratic derivatives,

of field theory and denoting

is generalized in order to embrace n first order partial differential

tions, and the theory is developed the number of independent

1.

used in the context

forms in the field variables and their first space and time partial using the idea of reciprocity

equa-

in order to ascertain

zilch densities that exist for such equations.

INTRODUCTION

Lipkin magnetic

[l] has developed field variables

the word zilch.

certain quadratic

in a relativistic

Four-dimensional

framework

vectors

forms are such that their four-divergence

quadratic density

and the

authors

have

Broer

to

energy

electromagnetic

extended

relations

of the number Heading systems tion

relating

these

stress

of simultaneous

being

of reciprocity,

analytical energy,

that

Kibble physical

which concerned

the concept

than

momentum

[2]

and other

situations,

and

waves, using the calculation

by the model under consideration. of zilch resides

linear partial differential rather

flux,

to plane electromagnetic

in his investigations

[4] has shown

tensor.

forms but are

of the more familiar

energy

to further

of zilch forms generated

quadratic

of the field variables,

density,

ideas

[3] has applied the concept

reciprocity

from these

These quadratic

of the first time and space derivatives forms

the electro-

for which he introduced

formed

vanishes.

involve the first time and space derivatives independent

forms involving

physical.

equations, The

in certain

his investiga-

generalized

and zilch have been shown to emerge

concepts only when

0 American Elsevier Publishing Company, Inc., 1973

J. HEADING

20 certain

conditions

defining

Heading’s which

are fulfilled

the linear system the

products.

investigations

reciprocity

flux

paper produces

Broer,

Maxwell’s

that

firstly point

equations,

and notation

by Lipkin.

of the tensor

properties

of

paper, the basic idea behind Broer’s

theory

zilch forms Secondly,

differential

appears,

Matrix theory is used,

showing how zilch arises solely from the

of view of the analytical

partial

the

rather than simplify it, since it gives

In the present

and how the resulting linear

analysis

from

into

from which Kibble’s

is modified and, indeed, simplified.

for Maxwell’s

originally

communication

are rearranged

using tensor analysis,

the situation

equations.

forms of zilch, in

as a sum of vector

a private

equations

zilch arises on account

equations.

communication

to special

all forms of zilch that may arise

The use of tensor

to complicate

roots of the matrix

differential

is represented

Maxwell’s

relation

forms of zilch arise. the impression

density

He acknowledges

in which

form of a reciprocity however,

were restricted

The present

from a linear system. Professor

by the characteristic

of first order partial

of partial

differential

are not independent

the theory

equations.

is extended

Maxwell’s

seen to be a very special simple case of a broadening

equations,

of those

given

to 1%first order

equations

can then

be

pyramid of generaliza-

tion. 2.

RECIPROCITY

FOR

MAXWELL’S

EQUATIONS

The notation curlwr

= -Y,

aw,

with div wr = div w2 = 0, is employed notation

enables

involved,

and we write these equations

where D E a/at. introduce

yielding

us to introduce

If

awl ,

curl we = ~

to permit generalization.

a linear

transformation

Matrix

of the vectors

in the form

T denotes a constant

the linear transformation

at

nonsingular

2

x

2 matrix,

we

GENERALIZED

The suffixes

ZILCH

JORDAN

CANONICAL

21

FORM

1, 1) are employed

(-

The characteristic responding

AND

vectors

roots (_f)

to facilitate generalization. 0 -1 o are + i, with corof the matrix 1 ( )

and(:).

Hence if’we choose

T= that is, Wl = z-1

=

z-1 +

Zl,

.$(\Vl + iw,),

iz_, + iz,,

IV2

=

-

z1

=

i(Wl -

iw,),

we obtain

namely curl z_1 = iDz_1, with div z_r

Dzr,

div zr = 0.

The equations

are no longer simultaneous

and the two pairs of equations

such as a column

in the two vector variables,

may be expressed

where a prime denotes the transpose.

Through

z and a corresponding

dual of z with respect to an orthogonal are related

curl zr = -

in the form

this investigation,

skewsymmetric

rotation

matrices

matrix

Z (the

of three-dimensional

axes)

by

Thus a vector product

u A v means the same thing as the column matrix

Uv, as may be checked

immediately.

22

J. HEADING

The two sets of Eqs. z_i may be distinct be written

(3) are independent,

in zi.

These equations

in

may

simply in the form

0'31= 0,

0'3-1= 0, where

so wi and w2 occurring

from wi and ws occurring

the

row

31,3_1are

operator

q’ denotes

skewsymmetric

matrices

(-

iD

(4) The

8’).

two matrices

of order four.

We note that

3-131=(_ y_, $)(fI since Z_,Z,

= z,z’,

product operator

31 commute.

-

-;')=(;I;:

being the expansion

(zllz,)I,

z_i A z, A. 53-r

Moreover,

z_lzll +:2zyy (a_,z,)I)J of the triple vector

has the same explicit form, so

the product

3_Ji

is symmetric,

3_1and

seen from the

above, or by noting that

(3-131)' = 31'3L (each is skewsymmetric)

= 3i3_i

3_131

=

(commutative

It is this fact that leads to the existence We now consider 0, appended

of reciprocity

the four-divergence

to a symbol

be differentiated

thus,

rule for differentiating

and zilch.

3_131.A

of the product

will denote

(3&,

by an operator

multiplication).

suffix

that the term is not to

placed on its left.

In keeping

with the

of two terms, we have

once a product

q'El311 = q'[3-l(3dOl + q'1(3-do311 = @'3-1)(31h + q'mlh311' (symmetric) = 0 + q'[31(3_1)01 = 0, by Eqs. (4). Hence the four-divergence this

is the required

different

reciprocity

fields wi, ws are used in 3-i

One real solution real solution Placing

of Eqs.

of the product 3-i&

relation

for Maxwell’s

vanishes, equations

and when

and 3i respectively.

(1) will be denoted

by xi, x2, and a second

by yi, y2. z_i = xi + ix,,

since all equations

z2 = yi -

are homogeneous),

its real and imaginary

parts thus :

iy,

(the factors

4 are irrelevant

we resolve the product

3_131into

GENERALIZED

ZILCH

AND JORDAN

CANONICAL

23

FORM

3-131 Xl’Yl + X2’Y2 =

iXlYl +i

(XlYl

+ X2Y2

X2’Yl ( X2Yl

X1Y1’ + Y1Xl’ +

(X2Yl

XlYZ

so the vanishing

reciprocity

X2Y2’ + YZX,’ -

Xl’Y2

-

+ X2Y2)’

X2Y1’ -

-

X1Y2’ + YlX2’ -

of its four-divergence

(-

@1’y1+

XlY2)’ Y2X1’ -

V’)

iD

x,‘Y,)I

(Xe’Y1 -

(3_131) yields

X,‘Y,P

) ’

the eight

relations D(x,‘Y,

+ XZ’YZ) -

W,‘Y,

-

WRY,

(X2’Yl -

W,Y,

-

-

-

Energy-momentum

+

V’[x2~1’

-

X,Y,)

= 0,

(5)

+ X,Y,)

= 0,

(6)

~1~2’ +

~1x2’

-

~2x1’

Xl'Y2)Il = 0,

X,Y,)’

(X1’Yl

-

~1’~s) + V’(X,Y,

+ X,Y,)’

-

V’(&YI

+

V’[xlyl’

X,‘Y,)~l

(7)

+ ~1x1’

+

~2~2’ +

YZA

= 0.

(8)

forms arise by placing xl = yl, x2 = ys;

and (7) vanish identically,

Eqs.

(6)

(5) and (8) yield (in vector notation

while Eqs.

as far as possible) $D(x12 + xz2) + V’(x, D(x,

A x2)’ -

V’[xlx,’

These equations

obviously

the momentum

density

A x2) = 0,

+ x2x2’ -

(9)

$(x1” + x~~)I] = 0.

contain the energy density, the Poynting

and the electromagnetic

It is evident that yi = Dx, in which case the reciprocity

E i,, yz = Dx, relations

PO) vector,

stress tensor. E f, also satisfy Eqs. (l),

yield

D(x, . i, + x2 - i2) + V’(x, A g2 -

x2 A il) = 0,

(11)

D(x, . i, - x1. i2) + V’(x, A ir, + x2 A X2) = 0,

(12)

D(x, A ii + x2 A ira)’ + V’ [Xi X2’ D(x,

A i,

-

x2i1’

+ 2,x,’

-

i,x,’

-

+ x,i,’

+ li2xg’ -

(x2. x1 -

x1. ir,)I] = 0, (13)

xl A g2)’

+ V’[X,%i’

+ i,x,’

(x1 * i,

+ x2 * $)I] = 0. (14)

24

J. HEADING

Obviously Eq.

Eq.

(11) is merely

(14) is the differential

Eqs.

(12) and

independent Broer satisfy

(13) reproduce

of Eqs.

certain

Differentiation

of Lipkin’s

a comma,

component

thus:

zilch forms

(9) and to time. that

are

yi = -

x2, YZ = xi, since these

zilch forms are produced

by this device.

with respect to x, y, or z will be denoted by

x, y, z respectively

,4 particular

of Eq.

(10) with respect

(11) and (14).

(l), but no further

Notation.

coefficient

of Eq.

[3] has used the substitutions

Eqs.

a suffix

the differential

coefficient

preceded

by a comma;

thus

axJay

= x~,~.

of a vector xi will be denoted by a suffix without

xiv, (x2 A x~)~. Hence a derivative

of a component

will

appear as xl?J,Z = I?,/&. It is evident that we may also substitute 91 =

yielding

forms similar

and x2 replaced 3. DEPENDENT

=

x2.z;

Yl = x1,%0

Yz =

%Y;

Yl = X1.U

Ya =

%!,z;

to Eqs.

Y2

(ll)-(14),

by derivatives

with the time derivatives

with respect

of equations

to sixteen

these represent

arises.

The four Eqs.

(11) and (14) are

when the x, y, and z-derivatives

all the first derivatives

are employed;

of the energy-momentum to sixteen

totally,

forms. yielding

forms of zilch.

To eliminate

any dependent

of zilch, it is sufficient densities

of xi

to X, y, z respectively.

The four Eqs. (12) and (13) are also augmented sixteen

relations

FORMS

A large number augmented

Xlm

into the reciprocity

will thereby

demanding

take care of themselves

at the most

The sixteen

forms existing

among these sixteen

to consider only the zilch densities. the extraction

zilch densities

in any linear combinations,

of irrelevant

solenoidal

vectors.

are xg *

x1

x1 -

x1.

k,,

(15)

x2

A X2,

(16)

-

Xl * x2.m

(1’)

-

Xl

(18)

A Ii, +

x2 * Xl x2

forms

The zilch flux

* x1.y

,z

* x2,9j,

GENERALIZED

ZILCH

AND

JORDAN

x2

to which for convenience

x l,z

*

verified

(19)

X2,?,

A x1x +

x‘2 A x2.m

Xl

A Xl,z/

+

x2

A x2,,,

(21)

XI

A x1.z

+

xz A x2.m

(22)

we shall append

vector

:

Xl.

~

25

FORM

Xl

(20)

the energy-momentum

(23)

A x;.

(24)

identity

for two vector

fields f and g is easily

f, divg + (f A fi,Jy - (f A a,,!, + (f A curld, - f. g,, = 0, and two further

similar

It is worth pointing subtracted

identities

out that

from Eq.

forms

B(Xl”+ xFz2), x1

The following

CANONICAL

obtained

by permuting

if f and g are interchanged

(25), we obtain

the x-component

(25)

the suffixes. and the result

of curl(f A g) in

the form

f,

-curl(f A g)], =

f

div g R . f,, -

This is, of course, identical [curl(f A g)], Formula

g, div f + (f A curl R)~ -

=

(25) represents,

(g A curl f)z

f . a.z.

with the more usual form

fz div

g - ,gTdiv f + (g . V)f, -

in effect,

a resolution

(f . V)g,.

of this last identity

into

two parts. If we first place f = g = xi and secondly f = g = x2 in (25), we obtain upon addition

the three identities

xlz div xi + x2, div x2 f

0(24),

-

(23),z = 0,

xly div x1 + xaY div xL?+ (20)z -

(22), ~ D(24),

-

(23),,

xiz div xi + xgz div xz + (21)p -

(20), -

where the numbers

(22), -

in parentheses

(al),

represent

-

D(24)z -

in Eq. (25) and subtract,

we obtain

(23),z = 0,

the equation

If now we first place f = x2, g = x1 and secondly

s 0,

so numbered.

f = x1 and g = x,

26

J. HEADING

xsZ div x1 -

xra: div x2 -

(IS),

-

(17) -

(24),,Z + (24),,,

= 0,

xsU div x1 -

xIV div xz -

(16), -

(18) -

(24),,,

+ (24),,,

= 0,

xgZ div x1 -

xIZ div x2 -

(16), -

(19) -

(24)2,y + (24),,,

= 0.

These six linear combinations zilch densities

reduce the possible number

of independent

to ten.

Moreover,

we note that

(20)= + (21), + (22), + x1 * curl x1 + x2 * curl x2 = 0, namely (20)s + (21), + (22)Z + (15) = 0, reducing

the number

The forms that

of independent

are more complicated more complicated equations

naturally

than Lipkin’s

forms.

to at most nine.

from reciprocity And Lipkin’s

relations

forms are also

than they need by the fact that a tensor

form of the

had been sought in order to see how the zilch densities

densities

transform

consideration, identity

zilch densities

we have derived

under

considerable

a Lorentz

for every variation

D [x,,(curl

transformation.

simplification of components

Apart

can be achieved

and flux from

this

by noticing

the

i and i:

x~)~ + x2j(curl x~)~] = D( - x~$.,~ + x~$~J

= - Xliji2j + X2jZli = - XliwYZi + X,jV%,, = since each xIi(curl

xJ3

component

sectionalization

of tne zilch densities given by Lipkin. linearly

in those derived

may be seen from the following

The x component, reexpressed

for example,

- x2jvx4,

V2$ = D2$.

the wave equation

+ xzj(curl x~)~ serves as a zilch density,

forms are included relations

satisfies

v . (XliVX,j

above

The fact that these from the reciprocity

argument.

of the zilch density

Eq.

as (XlhX =

%Xlr.z

l,z)z + -

(x2 AX2,r)r

XlPl2/,2 +

Hence

and forms, in fact, a

x2?+2z,z -

X2zG.fl.r

+

~21I~ZZ)~

(20) may be

GENERALIZED

an equation density. forms.

ZILCH

sufficient

Altogether,

AND

JORDAN

to show that densities

Missing combinations

these satisfy

CANONICAL

xlVxlZ,Z + xZU~Zz,eis a form of zilch

Eqs.

(20), (al),

and (22) yield eighteen

such

are forms such as xlIxIZ,Z + xa,~s+,~.

But

the trivial identities

there are in all nine such expressions. allowed for, and since xIi(curl we conclude

that Lipkin’s

-

= SC&S +

expressed

-

(X2v~2z),u

-

in terms

relations.

since -

is now

are embraced

linearly by

For example,

x&

&J,,

(%.~lz).a:

possible combination

x~)~ includes such expressions,

forms of zilch density

QV + xa,(curl

x21/,1/x2z-

Every

x~)~ + x,,(curl

those derived from the reciprocity xr,(curl

27

FORM

div x1 = 0

(~lV.1/~1* +

X2Y,YX2,) -

SC&,,

&,,,Z>

of part of (x1 A x~,~ + x2 A x~,~)~ and a three-diver-

gence. 4. THE GENERAL

Following

EQUATIONS

the previous paper by the writer

[2], we consider the n linear

equations curl wi = 2

aijDwj

(afj

real)

j=l

with the subsidiary

conditions

div wi -_ 0, for i = 1, 2,. . . , n.

tion to the Jordan

canonical

form with n vectors

variables

will yield

characteristic simultaneous

roots

n equations of the matrix

equations

aij is to be nonsingular

wholly aij

when repetitions

nonsimultaneous

are distinct,

The reduc-

zi as the transformed or may

when

n

yield simple

occur among the roots.

in this investigation.

the

Matrix

28

J. HEADING

In particular, root

-

we consider

4, such that

2 and .- q is, say, three. number

consistent

In Maxwell’s

the case when a root 2 occurs as well as a

the minimum with

equations,

of the two multiplicities

of the roots

We adopt the figure three as being the smallest perfect

clarity

the roots

for

immediate

are i and -

generalization.

i, so the figure

three is

reduced to one, and the analysis in this case gives no hint as to the process of generalization. We shall discuss

the case when the

Jordan

canonical

form implies

among others the equations curl 2-s =

qDz_, + Dz-,,

curl z_* =

qDz-,

+ Dzz,,

curl z-i =

qDz_,,

curl zi =

-

curl zs =

-

curl z3 = -

qDz,

qDz,

@z,, + Dz,,

+ Dz,,

namely a case when pure diagonalization

(26)

is not possible.

We are postulating

that at least one of ZQ or z4 must not occur associated or 4 respectively

in this set of simultaneous

we shall write these equations

equations.

In matrix

in the form:

,

0’3-3 = (- qD V’) ~1’3-2= (- qD

with the root -

0

-

z-3

-

z-3

q-12%2

+

gpz:z

7‘-3

8’)

0’3~ 3 (-

qD

V’)

0’3~ G (-

qD

V’)

0’3~ s (-

qD

V’)

0 ( Zl

-z1’

z, ) = 0

z, -

0, -

0

z3 - q-lzz

zz’ + q-121’ 7‘2

g-12, -

Z3’ + 4-12s’ Z3

= 0.

notation,

9

GENERALIZED

All the

3

ZILCH

matrices

JORDAN

CANONICAL

29

FORM

are defined to be skewsymmetric.

We have already that the product

AND

shown that the matrices

3r3_r

is symmetric,

yielding

3r and

3_1

commute,

the reciprocity

and

relation

q‘(313-I) = 0. This process may be generalized. It may easily be verified

that

0

(

313-Z - 323-l - 2q-l o

0

0

(

1

0

= 3-231 - 3432 - 2q-l 0 z_lzl’

&

i

’

0

0

313-3 - 323-2 + 333-l - 2 ( 0 q-yz1zL2 - z,z’_,) + q-2z1z:* 1 ’ 0

= 3-331 - 3-232 + 3-133 - 2 ( () q_yZ_eZ1, _ z_yz2,) + q_2z_1z1t1 . Moreover,

each of these left and right hand sides are symmetric

We now consider identities,

the four-divergence

using the operator

denote that an expression

q 0’).

(-

Again we use the suffix

is not to be differentiated.

0 to

We obtain

2q-f:

0’ [313_2 - 323-1 = 0’ 3,(3-do [

matrices.

of the left hand sides of these

- 32(3-do - 2q- l c:

z,(z~l)~l

~z,~:LI’ %?%I -3-1(32)0 - 2P-

+ 0’ (31)03-z - (32)03-I - 2q- l6 [

(271

= 0, since V’z, = V’z_, = 0. Similarly

33-3

This result is a second reciprocity

a third reciprocity

- 323-2 + 3334 - 2

relation

0

relation.

is given by

0

I 0 q-1(z1zL2 - z2z_1) + q-2z,zLl

I

=

0.

J. HEADING

30 Explicit purpose. involve (28).

evaluation

of these products

We shall consider the explicit

is not necessary

the reciprocity

matrices

densities

for our present

only, which do not

shown on the right of expressions

In order, these densities

(27) and

are zi’z-1, z,z-,,

Zi’Z-s -

z,z_,

zs’z-1 + 2q-lz,‘z_,,

- z,z_,

+ qpz,z_,,

Zl’Z_g - z2’z_2 + 23’2-1 + 2q-l(z,‘z_, z,z_, The structure

- z,z_,

+ z,z_,

- zz’z-1) + q-221’2-1,

+ q-yz,z_,

- Z,z_,).

of these forms shows that they may be combined

to yield the simple equivalent

reciprocity

densities

(in vector

linearly notation)

z1 * z-1,

z1 A z-1, z1 * z-2 - z2 * z-1, z1 A z-2 - z2 A z-1, Zl’ Z-Q - Q.‘Z_2 + z,hz_,

It

must be realized

densities.

that

Real and imaginary

when the complex operator be rearranged imaginary 5.

-

z2 AZ-,

Z3'Z_l,

+

z3

(29)

AZ-,.

these are complex

reciprocity

relations

and

parts can only be taken of the four divergence

(-

0’) is used. But these can immediately

qD

in order to yield densities

that

are merely

the real and

parts of forms (29).

GENERALIZED

ENERGY

AND

ZILCH

The canonical form for the matrix transformation

relating

transformation

as

A G (a,$) is achieved

the w’s to the z’s.

by a nonsingular

We shall write the inverse

n zv = c

b,,w,,

complex

numbers

q=1

where the b,, are in general

although

the aSj are real.

GENERALIZED

ZILCH

AND

Using the summation

CANONICAL

convention,

~1 =

where distinct

JORDAN

solutions

31

FORM

we may write

bi,x,,

z-1 = L,y,>

(39)

x, and yn are taken for the w,.

In the second place, we may also take ZI

=

z_~

by,,

Two forms of reciprocity

density

If in particular identical,

arise for

we choose ya =

x,,

these reciprocity

,v denotes

x, y or z respectively,

differentiation

independent

the difference

is skewsymmetric, of construction

(32). These definitions

we may substitute

blpb_-lrxa,v *

x,.

The sum of

and the number

-

of these two forms, W-&a

of

(34)

zilch density.

while that appearing ensures

that

namely

* xr.w

Eq.

The square

matrix

in

in Eq. (32) is symmetric;

(34) is independent

of Eq.

agree with the simpler ones adopted in our treatment

equations.

The same ideas hold for all twelve forms (29). yield the reciprocity Lb--lrxa density.

(32)

. XT,,.

forms has not been doubled by this device.

and we define this to be a complex

leading

is

(33)

(32), so they are not independent,

(b&i,

of Maxwell’s

t,

density

relations

Similarly

blnb_lrxp * x,,,, and secondly

We also consider

this method

to either

z-i = b_i,x,,

zi = bipxg,w,

z1 A z._~

become

z-1 = b--lpxq,v,

smce yp. E x~,~ satisfies the original equations.

firstly

+ V-&,

into the reciprocity

zi = bi,x,,

respect

of the energy

* x,,, = hb-1,

But we may also substitute

these forms is Eq.

with

then the first derivative

W-1rxq.v . x, + bd-+q

brackets

densities

and the result is known as an energy density, namely blab_lrxq - x,.

If the notation

yielding

(31) namely

z1 . z_~,

blPb--lrxT. yp.

and

bi&-irxa . yr

b-lpx,.

=

Thus the three forms

densities A ~7

and

bi,b-i,y,

A x7,

to blpb_lrxa A x, when yn G x,, a generalization The derivative

of momentum-

of this form is (b&i,

-

W--lJxu

A x,,,.

(35)

32

J. HEADING

But when we substitute

Addition

Eqs. (30) and (31), we obtain the densities

yields Eq. (35), while subtraction (b&-i,

an independent

+ bi+-ia)xg

A x,,,,

form that must also be described

When the operator

(-

gD

from the real and imaginary the vanishing

yields

as a zilch density.

V’) is applied, 4 being complex, parts

of the resulting

of the four-divergence,

parts of the complex zilch density

that

that

some of these contributions

the number

ninety-six

of real zilch

being produced

But the following matrix complex

number.

resolution.

certain

namely

A E (Q),

when

with the other forms

In the case of Maxwell’s but in general

is doubled

4, q, q, -

q, - q, -

q is complex,

since

the characteristic

it follows that q*, q*, q*, -

a second

associated Then

set of equations

the complex similar

another

whole set (29) is repeated Consider the matrix

complex

to set (26).

equation

Let

Hence

there

the variables

by 3-s, 3-s, 3-i, 3i, 3s, 3a.

reciprocity

density;

in fact,

the

with these new variables.

T that transforms

B into canonical

where

‘q10

J=

a

q*, - q*, - q* are

conjugate.

with this second set of roots be denoted

3i’ * 3_1 forms

of the

process may take place upon

denotes

exists

equation

q, where q is generally

also roots,

a star

it appears resolution,

show how this must be interpreted.

possesses real coefficients, where

by this

roots of the characteristic

If q is real, this doubling

But

by

from the set (29).

arguments

We are considering

vanish,

densities

it follows formed

both the real and imaginary

Eq. (34) (together

in set (29)) can be regarded as real zilch densities. equations,

equation

0 Q 1 0 0 4 ...

\

.**

... ...

-q

0

...

1

...

0

--4

0 0

1

--4j

form C, namely

GENERALIZED

ZILCH

AND

The first six columns

JORDAN

CANONICAL

in T are associated

33

FORM

with the first six columns

in

the identity

AT = TC, namely with the elements of J. The second six columns are associated with J*, and since A is real it follows that the second

six columns

are the conjugates

In the inverse

zi in terms of the wi (taken bij previously cofactors

of the first six columns

transformation,

used.

the elements

providing

the vectors

the coefficients

bij consist of the transposed

of T.

Let [T,; denote the column consisting of T.

respectively.

expresses

T-l

to be real), thereby

Essentially,

of the elements

the matrix

of the cofactors

of the jth column

Then

z_~ = [T,]‘w,

[T,]‘w,

z-3 =

Now the cofactors

appearing

perhaps a minus sign appended; the columns

in

[T,] and

[T,] are conjugates,

with

this may easily be checked by considering

used in the formation

[T3] and [T,] are conjugates,

3_3= [T,]‘w,. . . .

.. ,

of the cofactors

involved.

Similarly,

and so on.

Hence zl-rzl = w’[T,]‘[T,]w, and

31131 = w’[T,]‘[T,,]w It

follows

resolution

that

z:rzr

into

real

and 3Lr3r

a w’[T3*]‘[T4*]w.

are proportionally

and imaginary

parts

conjugates,

of equations

so the

involving

these

(and all) densities totally does not double the number of equations produced. The set of densities the corresponding set

(29) yields

independence

(29) yields forty-eight

conjugate

ninty-six,

while

or otherwise

complex

set yields a further the

conjugate

zilch relations,

forty-eight.

set is disregarded.

of this set of zilch equations

and

Otherwise, must

The now be

investigated. 6. DEPENDENT

AND INDEPENDENT

FORMS OF ZILCH

We may first consider the first four forms occurring the derived energy densities

zi . z- 1.2)+ z1.v. z- 1 f zrhz-

in set (29), yielding

and the zilch densities

qz1 . z_l)/av,

i,V + z1.w A z-1 = +1

A Z-l)/%

(36/J (37/V)

34

J. HEADING

Zl ’ z-1.v -

z1 A Ll,v there being sixteen

complex

-

Zl,vZ-1,

(38,J

z1.v AZ-,,

W/J

zilch densities

in forms (38) and (39).

We first place f = zi, g = z_i in identity g =

zl, where zi and z_, are both expressed

Bearing

in mind that the divergence

(25), and secondly

f = z_~,

in terms of the same fields w.

of all vectors z vanishes, we have upon

addition (39,,),

-

(39,,),

with two similar equations, (37/J, with two similar

-

+ 4(37/J,

-

(38,x) = 9,

(49)

and upon subtraction (37,,),

equations.

+ 4(39/,)x -

In general,

(38,,)

= 9,

(41)

twelve relations

the zilch forms when resolved into real and imaginary

exist

between

parts.

We also note the identities

(~1A z-dr

+ (zl A ~-~.~h,+ (zl A z-~.~)~= = -

(z-1A ZI.& + (x1 A zdv + (z-1A z~,~hAddition

qzl.

Dz-,,

z-1 * curl ~1 = qcl

* Dzl.

yields (39,e)z + (39,,),

and subtraction

the former alone referring relations.

+ 4(38/J - 9,

+ (37,&

+ q(36,,)

to zilch, implying

Hence the thirty-two

and (39) are reduced different

+ (39,,),

(42)

yields (37,,)0 + (37,,),

- 9,

generally

a further

real zilch densities derivable

at most to eighteen.

The conjugate

two real

from Eqs. (38) set leads to no

conclusions.

When

the pair of characteristic

case of Maxwell’s conjugate zi = bl,x,, sixteen

z1 * curl zF1

equations),

set exists.

For example,

z_~ = b_l,xr,

complex

roots

special

when q = f

is necessary

(as in the since no

icr (IX real), we note that

and under these circumstances

zilch densities

b_l, = b:,

so the

(38) and (39) become

(b,,b:- b,,bl*,)x, . x,,,, the first bracket

are pure imaginary

consideration

being pure imaginary

(h&i + b,,bl*,)x, A XT,,,, and the second

real.

There

are

GENERALIZED

therefore set.

ZILCH

AND

JORDAN

only sixteen real zilch densities,

The number

of independent

When CJ= f

The elements

reduces at most to nine.

essentially

contains

occurring

and their

respective

columns

from the other characteristic

must be either

of cofactors roots.

real or pure imaginary,

This means

the product

zi . z_~ must be either real or pure imaginary. there

nine independent

can be only sixteen

that

and it follows that

strates

that

conjugate

z1 = [T,]‘w,

consisting

each cofactor

arising

the cofactors

35

with no corresponding

densities

[T,]‘w,

of [T,],

in T, are such that conjugates

FORM

u (R real), we may take, as before z__~=

say.

CANONICAL

zilch densities,

This demon-

again reducing

to

forms.

We must finally the zilch densities

discuss the remaining

derived

therefrom.

reciprocity

densities

We may restrict

(29) and

the discussion

to

the forms zi

* z-

2.v

+

Zl,v

* z-62

-

z1 A z_L?.V+ zl,V A z-2 zr * zz1 A zthe latter

and combining. express

term

Eq.

2.v

-

-

z1.v

z1.u

equivalent

* z-1.v

-

z2.v.

z2 A z_l.v -

* z-2

-

z2 * z-1.u

A z-2

-

~2 A z-1.9

two forms giving sixteen

The identities

that

2.v

z2

complex

+

z2,v

z2.u

+

z-19

(43/V)

A z-1,

(44/J (45,lJ)

* z-1,

z2.u

(46/g)

A z-1,

zilch densities.

to Eqs. (40) and (41) are obtained

f = zi,

g = z-7,;

f = z-2,

g = z,;

f = z.2,

g = z-r;

f = z-1,

g=z,;

by placing

Since curl z_s and curl z2 now enter the identities,

these curls in terms

(37).

Totally,

of the time derivatives

we obtain

for the

identity

-

+ (37,,),

Eqs. (26)

yield an extra corresponding

to

Eq. (40): (46,,), The other

identities

-

(46~)s + q(44,& are similarly

modified,

unaltered

that at most only eighteen

contained

in the set Eqs.

In

all, the

densities conjugate

ninty-six

(43,,)

= 0.

but the conclusion

independent

remains

real zilch densities

are

(43)-(46). zilch

densities

(29) reduce to fifty-four set exists, the forty-eight

at most.

produced

from

the

reciprocity

In the special cases when no

zilch densities

reduce to twenty-seven,

36

J. HEADING

Considering

the whole set of characteristic

A, we may draw an overall conclusion. yields in the Jordan

canonical

form submatrices

there arise at most 1%~ or Qn independent

18n is used when there disregarded conjugate

(namely,

exists

zilch densities,

a conjugate

where

set, with the conjugate Qn is used

set

when

a

set does not exist (a = 0 or b = 0).

A possesses

at least

number of independent

exists only under the condition

one pair of characteristic

for every pair of submatrices

sections 4, 5 and 6 have shown how these zilch densities may be systematically

calculated.

REFERENCES

1 D. M. Lipkin, J. Math. B. Kibble,

Phys.

J. Math.

6(1964),

Phys.

3 L. J.. F. Broer, Physica 38(1968), 4 J. Heading,

Linear

Algebra

Received October 7977

roots

*

that the q.

The

zilch densities is at most equal to the sum of these

values of 1% or 9% respectively,

2 T. W.

& q that

of the form

q = a + ib, a # 0, b # 0) ;

Zilch derived from reciprocity matrix

roots of the original matrix

For every pair of roots

696-700.

6(1965),

1022-1026.

341-348.

2(1969),

413-425.

(47), and

and flux densities