Lectures on String Theory

– 10 –
are called the first class constraints. The mass-shell constraint for the relativistic
particle is of the first class.
There is another action for the relativistic particle.
characteristic features
It has the following two
• it does not contain square root
• it admits generalization to the massless case
Introduce e(τ) the auxiliary field on the world-line.
S = 1 2
∫ τ1
τ 0
dτ[ 1
e
( dx
µ
dτ
dx
) ]
µ
− em 2
dτ
Equations of motion:
for x µ d
( 1
)
eẋµ = 0
dτ
for e(τ) − 1
2e 2 ẋµ ẋ µ − 1 2 m2 = 0 =⇒ ẋ 2 + m 2 e 2 = 0
The last equation can be solved for e:
which leads to
d
dτ
e 2 = − 1 m 2 ẋ2
[
m
]
√ −ẋν ẋ ν ẋµ = 0
which is nothing else as the old eom for x µ . Also if we substitute solution for e into
the new action then this action reduces to the old one:
S = 1 ∫ τ1 [
√
m −ẋ
2 ] ∫ τ1
dτ √
2 −ẋ
2 ẋ2 −
m m2 = −m dτ √ −ẋ 2 (2.2)
τ 0
τ 0
Also
p µ = 1 eẋµ =⇒ p 2 = 1 e 2 ẋ2 = −m 2
but this time due to equations of motion for e. Equation of motion for e is purely
algebraic. The Hessian
∂ 2 L
∂ẋ µ ∂ẋ = 1 ∂
ν e ∂ẋ µ ẋν = 1 e η µν
is of maximal rank.
The constraint p 2 + m 2 = 0 does not follow from the definition of the canonical
momentum along, but one has to use equations of motion. Constraints which are
satisfied as consequences of equations of motion are called secondary.