y

x

z

S

y'

x'

z'

S'v

x = (x0, x1, x2, x3) (ct, x, y, z)

Relativistic (4-vector) Notation

compare: xi = (x1, x2, x3) = (x, y, z)

= = v/c 1

12

t' = (t-vx/c2)x' = (x-vt)y' = yz' = z

which can be recast as:

ct' = ctxx' = xv/c)ct

xo = xo x1

x1 = x1xo

which should be compared to the coordinate-mixing of rotations:

x' = xcos ysiny' = xcos ysin

Noticing:

2222 1

1

1

1cos

abba

a

212

1

/

22sin

ab

ab

ba

b

we write:

x0'x1'x2'x3'

compared to rotations about the z-axis:

x0'x1'x2'x3'

x0

x1

x2

x3

0 0

0 0

0 0 1 0

0 0 0 1

=

x0

x1

x2

x3

0 0

0 cos sin 0

0 sin cos 0

0 0 0 1

=

3

0

/

xx

exactly same form!

World line of particlemoving in straight linealong the x-direction

event

ct

xvt1

ct1

ct´

x´

The Lorentz transformation is not exactlya ROTATION, but mechanically like one.We will consider it a generalized rotation.

But now, the old “dot product” will no longer do.It can’t guarantee invariance for many of the

quantities we know should be invariant under suchtransformations. The fix is simple and obvious…

33221100 yxyxyxyxyx

So that under a Lorentz transformation

33221100 yxyxyxyxyx

3322

0101

1010

))((

))((

yxyx

yyxx

yyxx

3322

01012

10102

))((

))((

yxyx

yyxx

yyxx

3322

002100111

1120110002

])(

)([

yxyx

yxyxyxyx

yxyxyxyx

3322

2112002

)]1()1([

yxyx

yxyx

3322

2112002

)]1()1([)()(

yxyx

yxyxyx

21

1

33221100)()( yxyxyxyxyx

To help keep track of the sign conventions, we introduce the

metric tensor: 0 0

0 0

0 0 1 0

0 0 0 = g

Then our “dot product” becomes

yxygxygx

3

0

3

0

),,,( zyxctyy

and a lowered index means the metric tensor has been applied

Notice

yxyx

we argue:

ygx

yx

yxor

contra-variant4-vector

co-variant 4-vector

x

x

xx

xxx

summed over

The lowered index just meansthat is in the appropriateform to “dot” into a vector

Since the is raised, the above multiplication gives

(x´ )

Notice:

g means has been

multiplied by themetric tensor!

And just what does g look like?

1000

0100

00

00

1000

0100

0010

0001

g

1000

0100

00

00

1000

0100

00

00

g

Now notice that

(g) =

1000

0100

00

00

1000

0100

00

00

1000

0100

00)1(

00)1(2222

2222

1000

0100

0010

0001

= g

Also notice:

T = which means

gT =g gg or

but

(g)T = gThis is because:

(g)T = TgT = g = g

1000

0100

00

00

1000

0100

00

00

g

So as an exercise in using this notation let’s look at

)()( yx

)()( ygx

ygx

yxg

The indices indicate very specific matrix or vector components/elements. These are not matrices themselves, but just numbers, which we can reorder as we wish. We still have to respect the summations over repeated indices!

yxg??

(g) = gAnd remember we just showed

xyyx

i.e.

yxyx )()( All dot products are

INVARIANT underLorentz transformations.

yxg

even for ROTATIONS as an example, considerrotations about the z-axis

331212

212100

)sincos)(sincos(

)sincos)(sincos()()(

yxyyxx

yyxxyxyx

332111221222

222122121100

sincossincossincos

sincossincossincosyx

yxyxyxyx

yxyxyxyxyx

33221100 yxxxxxyx yx

The relativistic transformations:

)(c

Epp xx

)( xpc

E

c

E )( xcpEE

suggest a 4-vector

);( pc

Ep

that also transforms by

pp )(

so pp should be an invariant!

22

2

pc

E

Ec

In the particle’s rest frame:

px = ? E = ? pp = ?0 mc2 m2c2

In the “lab” frame:

)(c

Epp xx

)( xpc

E

c

E

)0( mc = mv

= = mc

2222222)()( cmcmpp

so

2222 )1( cm

22cm

Limitations of Schrödinger’s Equation

1-particle equation

),()(),(2

),( 21

22

txxVtxxm

txt

i

),,(),()()(

),,(2

),,(2

),,(

212121

2122

2

2

2

2121

2

2

2

21

txxxxVxVxV

txxxm

txxxm

txxt

i

2-particle equation:

mutual interaction

But in many high energy reactionsthe number of particles is not conserved!

np+e++e

n+p n+p+3

e+ p e+ p + 6 + 3