Lectures on String Theory

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which is in complete agreement with our previous discussion (We actually have to
identify θ = e). This Hamiltonian 3 should be provided with the constraint p 2 +m 2 = 0
which is the eom for e.
More generally, evolution of a singular system is governed by the Hamiltonian
H = H can + ∑ n
χ n φ n
Here {φ n } is an irreducible set of primary constraints and H can is the canonical
Hamiltonian:
H can = p µ ẋ µ − L
Only on the constraint surface φ n = 0 the Hamiltonian H coincides with H can . In
our present case
H can = m ẋνẋ ν
√ −ẋµ ẋ − (−m√ −ẋ µ µ ẋ µ ) = 0
and Hamiltonian dynamics of the system is due to the mass-shell condition only. The
choice of the coefficients χ n (τ) in H is equivalent to the choice of the gauge.
H = H can + χφ =
We get the time-evolution
θ
2m (p2 + m 2 ) , χ = θ
2m
dx µ
dτ = { θ
2m (p2 + m 2 ), x µ } = θ m pµ =
θẋ µ
√ −ẋµ ẋ µ
Therefore ẋ 2 = −θ 2 . Choosing θ = 1 means that we identify the time variable with
the proper time. This nicely illustrates the general point: in order to write down the
evolution equations in a system with local gauge invariance one has first to identify
the “time” variable.
Other gauge choices are possible. For instance, the static gauge consists in
imposing the condition t ≡ x 1 = τ. Equation for p t ≡ p 1 allows us to determine e:
δL
= dt
δp t dτ − ep t = 0 =⇒ e = 1 p t
The physical Hamiltonian dual to the world-line time τ coincides in this case with
the momentum p t conjugate to t: H = p t . It can be found from the eom for e:
p 2 = m 2 = 0 =⇒ − p 2 t + ⃗p 2 + m 2 = 0 =⇒ p t = √ ⃗p 2 + m 2 .
Note that here ⃗p = {p i } with i = 2, . . . , d.
case.
3 The gauge e = 1 m
is a close analogue of the conformal gauge to be considered for the string