Path integrals for actions that are not quadratic in their time derivatives

0.22 MB·English

by  Kevin Cahill

Path integrals for actions that are not quadratic in their time
derivatives
Kevin Cahill∗
Department of Physics & Astronomy, University of New Mexico,
Albuquerque, New Mexico 87131, USA and
School of Computational Sciences,
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1 Korea Institute for Advanced Study,
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2 Seoul 130-722, Korea

r
a (Dated: March 27, 2015)
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Abstract
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The standard way to construct a path integral is to use a Legendre transformation to find the

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hamiltonian, to repeatedly insert complete sets of states into the time-evolution operator, and then
t
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p to integrate over the momenta. This procedure is simple when the action is quadratic in its time
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derivatives, but in most other cases Legendre’s transformation is intractable, and the hamiltonian
[

3 isunknown. Thispapershowshowtoconstructpathintegralswhenonecan’tfindthehamiltonian
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3 because the first time derivatives of the fields occur in ways that make a Legendre transformation
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intractable; it focuses on scalar fields and does not discuss higher-derivative theories or those in
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which some fields lack time derivatives.
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∗ [e-mail]
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I. INTRODUCTION
Despite the success of renormalization, infinities remain a major problem in quantum
field theory, one that has grown more acute as cosmological observations have confirmed
the reality of dark energy [1]. For if dark energy is the energy of the vacuum, then we need
to be able to compute energies in finite theories. The ground-state energy of a theory with
hamiltonianH is thebigβ limit of−dlnTr[exp(−βH)]/dβ. We canstudy theground-state
energy of a theory if we can write the partition function Z(β) = Tr[exp(−βH)] as a path
integral in euclidian space.
If the action is quadratic in the first time derivatives of the fields, then the hamiltonian
is a simple Legendre transform of the Lagrange density, and we can use it to construct the
path integral in the standard manner [2]. But if the action is not quadratic in the time
derivatives of the fields, then the Legendre transform may be impossible, the hamiltonian
unknown, and the path integral a mystery.
Some quantum field theories have finite Green’s functions and finite euclidian action den-
sities [3, 4], but they have actions that are not quadratic in the time derivatives of the
fields. The Nambu-Got¯o action of string theory is not quadratic in the τ derivatives of
the fields Xµ(σ,τ). Apart from theories with unbroken supersymmetry, theories with fi-
nite energy densities tend to have actions that are not quadratic in the time derivatives of
the fields. The hamiltonians and path integrals of these theories are either complicated or
unknown. This paper shows how to construct path integrals when one can’t find the hamil-
tonian because the first time derivatives of the fields occur in ways that make a Legendre
transformation intractable; it focuses on scalar fields and does not discuss higher-derivative
theories [5] or those in which some fields lack time derivatives [6].
II. LAGRANGIANS AND HAMILTONIANS
The lagrangian of a theory tells us about symmetries and equations of motion, and the
hamiltonian tells us how to construct path integrals and how to compute the time evolution
of states and their energies. The standard way to construct a path integral is to use a
Legendre transformation to find the hamiltonian H from its lagrangian L and to insert
complete sets of eigenstates of the fields φ and of their conjugate momenta π into the
j j
2
time-evolution operator exp(−itH) so as to get
(cid:90) (cid:26)(cid:90) (cid:27)
(cid:104) (cid:105)
(cid:104)φ(cid:48)(cid:48)|e−itH|φ(cid:48)(cid:105) = exp i φ˙ π −H(φ,π)) d4x DφDπ (1)
j j
in which the time integral is from 0 to t, and the letters φ and π stand for all the fields
φ ,...,φ and momenta π ,...,π of the action [2]. If one can integrate over the momenta
1 n 1 n
and does so, then one has the usual expression
(cid:90) (cid:20)(cid:90) (cid:21)
(cid:104)φ(cid:48)(cid:48)|e−itH|φ(cid:48)(cid:105) = exp iL(φ,φ˙)d4x Dφ. (2)
This procedure is straightforward when the lagrangian is quadratic in its time derivatives,
but in most other theories the formulas that express the time derivatives in terms of the
fields and their momenta are insoluble and the hamiltonian is unknown. Most theories are
therefore inaccessible and unexamined.
The solution to this problem is to let functional integration perform Legendre’s trans-
formation (Sec. III) implicitly. Delta functionals can impose the relation between the time
derivatives and the fields and their momenta as illustrated by four examples in Sec. IV.
The cost is a doubling of the fields over which one must integrate and a determinant that
makes a ghostly appearance when it is positive (Sec. V). Similar delta functionals work in
euclidian space (Sec. VI). The Nambu-Goto action of string theory is not quadratic in its
time derivatives; its path integral is exhibited in Sec. VII.
III. THE LEGENDRE TRANSFORMATION
In theories of scalar fields, the momenta are derivatives of the action density
∂L
π = . (3)
j ˙
∂φ
j
˙
If one can invert these equations and write the time derivatives φ of the fields in terms of
j
the fields φ and their momenta π , then the energy density is
(cid:96) (cid:96)
n
(cid:88)
˙ ˙
H = π φ (φ,π)−L(φ,φ(φ,π)). (4)
j j
j=1
When the action is quadratic in all the time derivatives, this Legendre transformation is
easy to do, but in most other cases no solution is known even in the absence of constraints.
3
IV. A PATH-INTEGRAL LEGENDRE TRANSFORMATION
Let the field χ˙(φ,π) be the function of the fields φ and their conjugate momenta π that
j j
satisfies Legendre’s relation
∂L(φ,χ˙)
π = . (5)
j
∂χ˙
j
In terms of this in-general-inaccessible function χ˙(φ,π), the energy density is
n
(cid:88)
H(φ,π) = π χ˙ (φ,π)−L(φ,χ˙(φ,π)) (6)
j j
j=1
and the path integral (1) is
(cid:90) (cid:26)(cid:90) (cid:27)
(cid:104) (cid:105)
(cid:104)φ(cid:48)(cid:48)|e−itH|φ(cid:48)(cid:105) = exp i φ˙ π −(π χ˙ (φ,π)−L(φ,χ˙(φ,π))) d4x DφDπ. (7)
j j k k
Although we don’t know what χ˙(φ,π) is, we still can write this path integral in terms of a
delta functional as
(cid:90) (cid:26)(cid:90) (cid:27)
(cid:104) (cid:105)
(cid:104)φ(cid:48)(cid:48)|e−itH|φ(cid:48)(cid:105) = exp i φ˙ π −(π χ˙ (φ,π)−L(φ,χ˙(φ,π))) d4x
j j k k
(8)
(cid:89)(cid:104) (cid:16) (cid:17)(cid:105)
˙ ˙
× δ χ˙ (φ,π)−ψ DφDπDψ.
(cid:96) (cid:96)
(cid:96),x
˙
We can use the delta functional to replace χ˙ by ψ everywhere else:
(cid:90) (cid:26)(cid:90) (cid:27)
(cid:104) (cid:16) (cid:17)(cid:105)
(cid:104)φ(cid:48)(cid:48)|e−itH|φ(cid:48)(cid:105) = exp i φ˙ π − π ψ˙ (φ,π)−L(φ,ψ˙(φ,π)) d4x
j j k k
(9)
(cid:89)(cid:104) (cid:16) (cid:17)(cid:105)
˙ ˙
× δ χ˙ (φ,π)−ψ DφDπDψ.
(cid:96) (cid:96)
(cid:96),x
˙ ˙
The delta functional δ(χ˙ − ψ) imposes the relation (5) among the fields φ, π, and ψ. We
now use the delta-function identity
(cid:34) (cid:32) (cid:33)(cid:35)
(cid:89)(cid:104) (cid:16) ˙ (cid:17)(cid:105) (cid:89) ∂L(φ,ψ˙) (cid:12)(cid:12) (cid:18)∂πk(cid:19)(cid:12)(cid:12)
δ χ˙ (φ,π)−ψ = δ π − (cid:12)det (cid:12)
(cid:96) (cid:96) (cid:96) ∂ψ˙ (cid:12) ∂ψ˙ (cid:12)
(cid:96),x (cid:96),x (cid:96) (cid:96)
(10)
(cid:34) (cid:32) (cid:33)(cid:35)(cid:12) (cid:32) (cid:33)(cid:12)
(cid:89) ∂L(φ,ψ˙) (cid:12) ∂2L(φ,ψ˙) (cid:12)
= δ π − (cid:12)det (cid:12)
(cid:96) ˙ (cid:12) ˙ ˙ (cid:12)
(cid:96),x ∂ψ(cid:96) (cid:12) ∂ψk∂ψ(cid:96) (cid:12)
to change variables in the delta functional introducing the appropriate jacobian
(cid:90) (cid:26)(cid:90) (cid:27)
(cid:104) (cid:16) (cid:17)(cid:105)
(cid:104)φ(cid:48)(cid:48)|e−itH|φ(cid:48)(cid:105) = exp i φ˙ π − π ψ˙ −L(φ,ψ˙) d4x
j j k k
(cid:34) (cid:32) (cid:33)(cid:35)(cid:12) (cid:32) (cid:33)(cid:12) (11)
(cid:89) ∂L(φ,ψ˙) (cid:12) ∂2L(φ,ψ˙) (cid:12)
× δ π − (cid:12)det (cid:12)DφDπDψ˙.
(cid:96) ˙ (cid:12) ˙ ˙ (cid:12)
(cid:96),x ∂ψ(cid:96) (cid:12) ∂ψk∂ψ(cid:96) (cid:12)
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The repeated indices j,k,(cid:96) are summed from 1 to n. The integration is over all fields that
go from φ(cid:48) to φ(cid:48)(cid:48) in time t and over all π and ψ˙ in that time interval. Integrating over π, we
find
(cid:34) (cid:35)(cid:12) (cid:32) (cid:33)(cid:12)
(cid:90) (cid:90) ∂L(φ,ψ˙) (cid:12) ∂2L(φ,ψ˙) (cid:12)
(cid:104)φ(cid:48)(cid:48)|e−itH|φ(cid:48)(cid:105) = exp i(φ˙ −ψ˙ ) +iL(φ,ψ˙)d4x (cid:12)det (cid:12)DφDψ˙.
(cid:96) (cid:96) ˙ (cid:12) ˙ ˙ (cid:12)
∂ψ(cid:96) (cid:12) ∂ψk∂ψ(cid:96) (cid:12)
(12)
This functional integral generalizes the path integral to theories of scalar fields in which the
action is not quadratic in the time derivatives of the fields. A similar formula should work
in theories of vector and tensor fields, apart from the issue of constraints.
Our first example is a free scalar field with action density
1 1
L = ∂ φ∂µφ− m2φ2 (13)
µ
2 2
and canonical momentum
∂L
˙
π = = φ. (14)
˙
∂φ
The determinant is unity, and the proposed path integral (12) is
(cid:90) (cid:26)(cid:90) (cid:27)
(cid:104) (cid:105)
(cid:104)φ(cid:48)(cid:48)|e−itH|φ(cid:48)(cid:105) = exp i L(φ,ψ˙)+ψ˙(φ˙ −ψ˙) d4x DφDψ˙
(15)
(cid:90) (cid:26)(cid:90) (cid:20) (cid:21) (cid:27)
1 1 1
= exp i ψ˙2 − (∇φ)2 − m2φ2 +ψ˙(φ˙ −ψ˙) d4x DφDψ˙.
2 2 2
˙ ˙ ˙ ˙
Shifting ψ to ψ +φ and integrating over ψ, we get the usual result
(cid:90) (cid:26)(cid:90) (cid:20) (cid:21) (cid:27)
1 1 1
(cid:104)φ(cid:48)(cid:48)|e−itH|φ(cid:48)(cid:105) = exp i (ψ˙ +φ˙)2 − (∇φ)2 − m2φ2 −(ψ˙ +φ˙)ψ˙) d4x DφDψ˙
2 2 2
(cid:90) (cid:26)(cid:90) (cid:20) (cid:21) (cid:27)
1 1 1 1
= exp i φ˙2 − (∇φ)2 − m2φ2 − ψ˙2 d4x DφDψ˙
2 2 2 2
(cid:90) (cid:26)(cid:90) (cid:20) (cid:21) (cid:27)
1 1 1
= exp i φ˙2 − (∇φ)2 − m2φ2 d4x Dφ
2 2 2
(cid:90) (cid:20) (cid:90) (cid:21)
= exp i L(φ,φ˙)d4x Dφ.
(16)
Our second example is a field theory in one dimension, time. The lagrangian for a
relativistic particle of mass m
(cid:112)
L = −m 1−q˙2 (17)
5
is not quadratic the first time derivative q˙. This theory is special in that we can invert the
definition of the momentum
∂L mq˙
p = = , (18)
(cid:112)
∂q˙ 1−q˙2
express q˙ in terms of p
p
q˙ = , (19)
(cid:112)
p2 +m2
and find the hamiltonian
(cid:112)
H = p2 +m2. (20)
The standard double path integral (1) then is
(cid:90) (cid:20)(cid:90) (cid:21)
(cid:16) (cid:112) (cid:17)
(cid:104)q(cid:48)(cid:48)|e−itH|q(cid:48)(cid:105) = exp i q˙p− p2 +m2 dt DqDp. (21)
Suppose, however, that we were unable to perform Legendre’s transformation and get
these equations (19–21). In that case, we could use the proposed path integral (12)
(cid:90) (cid:20)(cid:90) ∂L(q,s˙) (cid:21)(cid:12)(cid:12)∂2L(q,s˙)(cid:12)(cid:12)
(cid:104)q(cid:48)(cid:48)|e−itH|q(cid:48)(cid:105) = exp i(q˙−s˙) +iL(q,s˙) (cid:12) (cid:12)DqDs˙. (22)
∂s˙ (cid:12) ∂s˙2 (cid:12)
We then would write
(cid:90) (cid:20)(cid:90) ms˙ √ (cid:21) m
(cid:104)q(cid:48)(cid:48)|e−itH|q(cid:48)(cid:105) = exp i(q˙−s˙) √ −im 1−s˙2 DqDs˙. (23)
1−s˙2 (1−s˙2)3/2
To check that this path integral (23) is the same as the standard double path integral
(21), we change variables in it (23), setting
p
s˙ = (24)
(cid:112)
p2 +m2
so that
m2
ds˙ = dp. (25)
(p2 +m2)3/2
We then find that
m (cid:18)p2 +m2(cid:19)3/2 m2
ds˙ = m dp = dp, (26)
(1−s˙2)3/2 m2 (p2 +m2)3/2
and that
ms˙ √ p2 m2
(q˙−s˙) √ −m 1−s˙2 = q˙p− −
(cid:112) (cid:112)
1−s˙2 p2 +m2 p2 +m2
(27)
(cid:112)
= q˙p− p2 +m2.
6
Thus the proposed path integral (12) reduces in this quantum-mechanical example to a path
integral (22) that is the same as the standard double path integral (21) for this example.
Our third example is the scalar Born-Infeld theory with action density
(cid:114)
(cid:16) (cid:17)
L = −M4 1−M−4 φ˙2 −(∇φ)2 −m2φ2 (28)
which is the field-theory version of the second example. The proposed path integral (12) is
(cid:40) (cid:34) (cid:35) (cid:41)(cid:12) (cid:12)
(cid:90) (cid:90) ∂L(φ,ψ˙) (cid:12)∂2L(φ,ψ˙)(cid:12)
(cid:104)φ(cid:48)(cid:48)|e−itH|φ(cid:48)(cid:105) = exp i (φ˙ −ψ˙) +L(φ,ψ˙) d4x (cid:12) (cid:12)DφDψ˙ (29)
∂ψ˙ (cid:12)(cid:12) ∂ψ˙2 (cid:12)(cid:12)
in which the partial derivatives are
˙ ˙
∂L(φ,ψ) ψ
˙
π(φ,ψ) = = (30)
∂ψ˙ (cid:114) (cid:16) (cid:17)
1−M−4 ψ˙2 −(∇ψ)2 −m2φ2
and
∂π(φ,ψ˙) ∂2L(φ,ψ˙) 1+M−4((∇ψ)2 +m2φ2)
= = . (31)
∂ψ˙ ∂ψ˙2 (cid:114) (cid:16) (cid:17)
1−M−4 ψ˙2 −(∇ψ)2 −m2φ2
Substituting these formulas into the proposed path integral (29) gives
(cid:90) (cid:26)(cid:90) (cid:104)(cid:16) (cid:17) (cid:105) (cid:27) ∂π(φ,ψ˙)
(cid:104)φ(cid:48)(cid:48)|e−itH|φ(cid:48)(cid:105) = exp i φ˙ −ψ˙ π(φ,ψ˙)+L(φ,ψ˙) d4x DφDψ˙. (32)
˙
∂ψ
˙
This theory is one of the few in which we can solve (30) for the time derivative ψ
ψ˙ = √ π (cid:112)1+M−4((∇φ)2 +m2φ2) (33)
1+M−4π2
and find as the hamiltonian density
˙ ˙ ˙ ˙
H(φ,π(φ,ψ)) = π(φ,ψ)ψ −L(φ,ψ)
(34)
(cid:112)
= (M4 +π2)(M4 +(∇φ)2 +m2φ2).
Thus the proposed path integral (32) is the standard formula (1)
(cid:90) (cid:26)(cid:90) (cid:27)
(cid:104) (cid:105) ∂π
(cid:104)φ(cid:48)(cid:48)|e−itH|φ(cid:48)(cid:105) = exp i φ˙π(φ,ψ˙)−H(φ,π(φ,ψ˙)) d4x DφDψ˙
˙
∂ψ
(cid:90) (cid:26)(cid:90) (cid:27)
(cid:104) (cid:105)
= exp i φ˙π −H(φ,π) d4x DφDπ (35)
(cid:90) (cid:26)(cid:90) (cid:27)
= exp i(cid:104)φ˙π −(cid:112)(M4 +π2)(M4 +(∇φ)2 +m2φ2)(cid:105)d4x DφDπ.
7
Our fourth example is the theory defined by the action density
L = M4exp(L /M4) (36)
0
in which L is the action density (13) of the free field. The derivatives of L are
0
∂L ∂2L
= M−4ψ˙ L and = M−4(1+M−4ψ˙2)L. (37)
∂ψ˙ ∂ψ˙2
So the proposed path integral is
(cid:40) (cid:34) (cid:35) (cid:41)(cid:12) (cid:12)
(cid:90) (cid:90) ∂L(φ,ψ˙) (cid:12)∂2L(φ,ψ˙)(cid:12)
(cid:104)φ(cid:48)(cid:48)|e−itH|φ(cid:48)(cid:105) = exp i L(φ,ψ˙)+ (φ˙ −ψ˙) d4x (cid:12) (cid:12)DφDψ˙
∂ψ˙ (cid:12)(cid:12) ∂ψ˙2 (cid:12)(cid:12)
(cid:40) (cid:34) (cid:35) (cid:41)
(cid:90) (cid:90) ψ˙(φ˙ −ψ˙) (1+M−4ψ˙2)L(φ,ψ˙)
= exp i 1+ L(φ,ψ˙)d4x DφDψ˙.
M4 M4
(38)
V. HIDDEN FERMIONIC VARIABLES
A determinant is a gaussian integral
(cid:90) n
(cid:89)
e−θ†Aθ dθ∗dθ = detA (39)
k k
k=1
as is well-known [2]. When the determinant is positive, we can drop the absolute-value signs
and write the proposed path integral (12) as
(cid:40) (cid:34) (cid:35) (cid:41)
(cid:90) (cid:90) ∂L(φ,ψ˙) (cid:18) ∂2L (cid:19)
(cid:104)φ(cid:48)(cid:48)|e−itH|φ(cid:48)(cid:105) = exp i (φ˙ −ψ˙ ) +L(φ,ψ˙) d4x det DφDψ˙
(cid:96) (cid:96) ˙ ˙ ˙
∂ψ ∂ψ ∂ψ
(cid:96) k (cid:96)
(cid:90) (cid:26)(cid:90) ∂L ∂2L (cid:27)
= exp i(φ˙ −ψ˙ ) +iL−χ¯ χ d4x DφDψ˙Dχ¯Dχ
(cid:96) (cid:96) ˙ k ˙ ˙ (cid:96)
∂ψ ∂ψ ∂ψ
(cid:96) k (cid:96)
(40)
in which φ and ψ are boson fields, and χ are scalar Grassmann fields or ghosts.
(cid:96) (cid:96) k
VI. EUCLIDIAN SPACE
It is easier to evaluate path integrals in euclidian space where in the proposed path
integral (12) the integral over x4 runs from 0 to the inverse temperature β = 1/kT
(cid:34) (cid:35)(cid:12) (cid:32) (cid:33)(cid:12)
(cid:90) (cid:90) (cid:16) (cid:17) ∂L(φ,ψ˙) (cid:12) ∂2L(φ,ψ˙) (cid:12)
(cid:104)φ(cid:48)(cid:48)|e−βH|φ(cid:48)(cid:105) = exp iφ˙ −ψ˙ +L(φ,ψ˙)d4x (cid:12)det (cid:12)DφDψ˙.
j j ˙ (cid:12) ˙ ˙ (cid:12)
∂ψj (cid:12) ∂ψk∂ψ(cid:96) (cid:12)
(41)
8
In this theory, the mean value of an observable A[φ] in a system at maximum entropy
and inverse temperature β is
(cid:34) (cid:35)
(cid:104)A[φ](cid:105) = TrA[φ]e−βH = (cid:90) A[φ]exp (cid:90) (cid:16)iφ˙ −ψ˙ (cid:17) ∂L +Ld4x (cid:12)(cid:12)(cid:12)det(cid:18) ∂2L (cid:19)(cid:12)(cid:12)(cid:12)DφDψ˙
Tre−βH j j ∂ψ˙ (cid:12) ∂ψ˙ ∂ψ˙ (cid:12)
j k (cid:96)
(cid:44) (cid:34) (cid:35)
(cid:90) exp (cid:90) (cid:16)iφ˙ −ψ˙ (cid:17) ∂L +Ld4x (cid:12)(cid:12)(cid:12)det(cid:18) ∂2L (cid:19)(cid:12)(cid:12)(cid:12) DφDψ˙ .
j j ∂ψ˙ (cid:12) ∂ψ˙ ∂ψ˙ (cid:12)
j k (cid:96)
(42)
This ratio of path integrals is a ratio of mean values
(cid:42) (cid:34) (cid:35)(cid:43)(cid:44)(cid:42) (cid:34) (cid:35)(cid:43)
(cid:90) (cid:90)
∂L ∂L
(cid:104)A[φ](cid:105) = A[φ]exp iφ˙ d4x exp iφ˙ d4x (43)
j ˙ j ˙
∂ψ ∂ψ
j j
each estimated by Monte Carlo simulation [7] in the normalized probability distribution
(cid:34) (cid:32) (cid:33) (cid:35)
P(φ,ψ˙) = exp (cid:90) L(φ,ψ˙)−ψ˙ ∂L d4x (cid:12)(cid:12)(cid:12)det(cid:18) ∂2L (cid:19)(cid:12)(cid:12)(cid:12)
j ∂ψ˙ (cid:12) ∂ψ˙ ∂ψ˙ (cid:12)
j k (cid:96)
(44)
(cid:44) (cid:34) (cid:32) (cid:33) (cid:35)
(cid:90) exp (cid:90) L(φ,ψ˙)−ψ˙ ∂L d4x (cid:12)(cid:12)(cid:12)det(cid:18) ∂2L (cid:19)(cid:12)(cid:12)(cid:12)DφDψ˙ .
j ∂ψ˙ (cid:12) ∂ψ˙ ∂ψ˙ (cid:12)
j k (cid:96)
VII. STRINGS
The tau or time derivatives of the coordinate fields Xµ in the Nambu-Goto¯ Lagrange
density
T (cid:90) σ1(cid:114)(cid:16) (cid:17)2 (cid:16) (cid:17)2
L = − 0 X˙ ·X(cid:48) − X˙ (X(cid:48))2 (45)
c
0
do not occur quadratically. The momenta are
∂L T (X˙ ·X(cid:48))X(cid:48) −(X(cid:48))2X˙
Pτ = = − 0 µ µ (46)
µ ∂X˙µ c (cid:114)(cid:16) (cid:17)2 (cid:16) (cid:17)2
X˙ ·X(cid:48) − X˙ (X(cid:48))2
and the second derivatives of the Lagrange density are [8]

∂2L T0  δµνX(cid:48)2 −Xµ(cid:48)Xν(cid:48)
∂X˙µ∂X˙ν = c (cid:114)(cid:16) (cid:17)2 (cid:16) (cid:17)2
X˙ ·X(cid:48) − X˙ (X(cid:48))2
 (47)
(cid:16) (cid:17)(cid:16) (cid:17)
(X˙ ·X(cid:48))X(cid:48) −(X(cid:48))2X˙ (X˙ ·X(cid:48))X(cid:48) −(X(cid:48))2X˙
µ µ ν ν 
− .
(cid:20)(cid:16)X˙ ·X(cid:48)(cid:17)2 −(cid:16)X˙(cid:17)2(X(cid:48))2(cid:21)3/2 
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The proposed path integral (12) for the Nambu-Goto action is then
(cid:34) (cid:35)
(cid:90) (cid:90) ˙
∂L(X,Y)
(cid:104)X(cid:48)(cid:48)|e−iτH|X(cid:48)(cid:105) = exp i(X˙µ −Y˙ µ) +iL(X,Y˙ )d4x
∂Y˙ µ
(48)
(cid:12) (cid:34) (cid:35)(cid:12)
(cid:12) ∂2L(X,Y˙ ) (cid:12)
×(cid:12)det (cid:12)DXDY˙
(cid:12)(cid:12) ∂Y˙ µ∂Y˙ ν (cid:12)(cid:12)
in which the formulas (46) and (47) (with X˙µ → Y˙ µ) are to be substituted for the first and
second derivatives of the action density L with respect to the tau derivatives Y˙ µ.
ACKNOWLEDGMENTS
ConversationswithDavidAmdahl, MichaelGrady, SudhakarPrasad, andShashankShal-
gar advanced this work which arose out of numerical computations done with David Amdahl
at the Korea Institute for Advanced Study and at the National Energy Research Scientific
Computing Center, a DOE Office of Science User Facility supported by the Office of Science
of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231.
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K. Cahill, Physical Mathematics (Cambridge University Press, 2013) Chap. 16.
[3] S. Boettcher and C. Bender, J. Math. Phys. 31(11), 2579 (1990).
[4] K. Cahill, Phys. Rev. D 87, 065024 (2013); Phys. Rev. D 88, 125014 (2013).
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304018 (2008).
[6] P. A. M. Dirac, Can. J. Math. 2, 129 (1950); Proc. Roy. Soc. London, ser. A 246, 326 (1958);
Lectures on Quantum Mechanics (Yeshiva University and Dover Publications, 1964).
[7] K. Cahill, Physical Mathematics (Cambridge University Press, 2013) Chap. 14.
[8] K. Cahill, Physical Mathematics (Cambridge University Press, 2013) Chap. 19.
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