Path integral in quantum mechanics Path integral formulation

1 path integral in quantum mechanics

1.1 time-slicing derivation
1.2 path integral formula
1.3 free particle
1.4 simple harmonic oscillator
1.5 coulomb potential
1.6 schrödinger equation
1.7 equations of motion
1.8 stationary-phase approximation
1.9 canonical commutation relations
1.10 particle in curved space
1.11 measure-theoretic factors

path integral in quantum mechanics
time-slicing derivation

one common approach deriving path integral formula divide time interval small pieces. once done, trotter product formula tells noncommutativity of kinetic , potential energy operators can ignored.

for particle in smooth potential, path integral approximated zigzag paths, in 1 dimension product of ordinary integrals. motion of particle position xa @ time ta xb @ time tb, time sequence

t

a


=

t

0


<

t

1


<
⋯
<

t

n
−
1


<

t

n


<

t

n
+
1


=

t

b




{\displaystyle t_{a}=t_{0}<t_{1}<\cdots <t_{n-1}<t_{n}<t_{n+1}=t_{b}}

can divided n + 1 smaller segments tj − tj − 1, j = 1, ..., n + 1, of fixed duration

ε
=
Δ
t
=




t

b


−

t

a




n
+
1



.


{\displaystyle \varepsilon =\delta t={\frac {t_{b}-t_{a}}{n+1}}.}

this process called time-slicing.

an approximation path integral can computed proportional to

∫

−
∞


+
∞


⋯

∫

−
∞


+
∞


exp
⁡

(


i
ℏ



∫


t

a





t

b




l


(


x
(
t
)
,
v
(
t
)


)



d
t
)


d

x

0



⋯

d

x

n


,


{\displaystyle \int \limits _{-\infty }^{+\infty }\cdots \int \limits _{-\infty }^{+\infty }\exp \left({\frac {i}{\hbar }}\int _{t_{a}}^{t_{b}}l{\big (}x(t),v(t){\big )}\,dt\right)\,dx_{0}\,\cdots \,dx_{n},}

where l(x, v) lagrangian of one-dimensional system position variable x(t) , velocity v = ẋ(t) considered (see below), , dxj corresponds position @ jth time step, if time integral approximated sum of n terms.

in limit n → ∞, becomes functional integral, which, apart nonessential factor, directly product of probability amplitudes ⟨xb, tb|xa, ta⟩ (more precisely, since 1 must work continuous spectrum, respective densities) find quantum mechanical particle @ ta in initial state xa , @ tb in final state xb.

actually l classical lagrangian of one-dimensional system considered,

l
(
x
,



x
˙



)
=
t
−
v
=


1
2


m

|




x
˙





|


2


−
v
(
x
)


{\displaystyle l(x,{\dot {x}})=t-v={\frac {1}{2}}m|{\dot {x}}|^{2}-v(x)}

and abovementioned zigzagging corresponds appearance of terms

exp
⁡

(


i
ℏ


ε

∑

j
=
1


n
+
1


l

(




x
~




j


,




x

j


−

x

j
−
1



ε


,
j
)

)



{\displaystyle \exp \left({\frac {i}{\hbar }}\varepsilon \sum _{j=1}^{n+1}l\left({\tilde {x}}_{j},{\frac {x_{j}-x_{j-1}}{\varepsilon }},j\right)\right)}

in riemann sum approximating time integral, integrated on x1 xn integration measure dx1...dxn, x̃j arbitrary value of interval corresponding j, e.g. center, xj + xj−1/2.

thus, in contrast classical mechanics, not stationary path contribute, virtual paths between initial , final point contribute.

path integral formula

in terms of wave function in position representation, path integral formula reads follows:

ψ
(
x
,
t
)
=


1
z



∫


x

(
0
)
=
x



e

i
s
(

x

,




x

˙



)



ψ

0


(

x

(
t
)
)



d



x




{\displaystyle \psi (x,t)={\frac {1}{z}}\int _{\mathbf {x} (0)=x}e^{is(\mathbf {x} ,{\dot {\mathbf {x} }})}\psi _{0}(\mathbf {x} (t))\,{\mathcal {d}}\mathbf {x} \,}

where





d



x



{\displaystyle {\mathcal {d}}\mathbf {x} }

denotes integration on paths




x



{\displaystyle \mathbf {x} }






x

(
0
)
=
x


{\displaystyle \mathbf {x} (0)=x}

,



z


{\displaystyle z}

normalization factor. here



s


{\displaystyle s}

action, given by

s
(

x

,




x

˙



)
=
∫
l
(

x

(
t
)
,




x

˙



(
t
)
)

d
t


{\displaystyle s(\mathbf {x} ,{\dot {\mathbf {x} }})=\int l(\mathbf {x} (t),{\dot {\mathbf {x} }}(t))\,dt}





the diagram shows contribution path integral of free particle set of paths.


free particle

the path integral representation gives quantum amplitude go point x point y integral on paths. free-particle action (for simplicity let m = 1, ħ = 1)

s
=
∫






x
˙




2


2



d
t
,


{\displaystyle s=\int {\frac {{\dot {x}}^{2}}{2}}\,dt,}

the integral can evaluated explicitly.

to this, convenient start without factor in exponential, large deviations suppressed small numbers, not cancelling oscillatory contributions:

k
(
x
−
y
;
t
)
=

∫

x
(
0
)
=
x


x
(
t
)
=
y


exp
⁡

(
−

∫

0


t








x
˙




2


2



d
t
)


d
x
.


{\displaystyle k(x-y;t)=\int _{x(0)=x}^{x(t)=y}\exp \left(-\int _{0}^{t}{\frac {{\dot {x}}^{2}}{2}}\,dt\right)\,dx.}

splitting integral time slices:

k
(
x
,
y
;
t
)
=

∫

x
(
0
)
=
x


x
(
t
)
=
y



∏

t


exp
⁡

(
−



1
2





(



x
(
t
+
ε
)
−
x
(
t
)

ε


)


2


ε
)


d
x
,


{\displaystyle k(x,y;t)=\int _{x(0)=x}^{x(t)=y}\prod _{t}\exp \left(-{\tfrac {1}{2}}\left({\frac {x(t+\varepsilon )-x(t)}{\varepsilon }}\right)^{2}\varepsilon \right)\,dx,}

where dx interpreted finite collection of integrations @ each integer multiple of ε. each factor in product gaussian function of x(t + ε) centered @ x(t) variance ε. multiple integrals repeated convolution of gaussian gε copies of @ adjacent times:

k
(
x
−
y
;
t
)
=

g

ε


∗

g

ε


∗
⋯
∗

g

ε


,


{\displaystyle k(x-y;t)=g_{\varepsilon }*g_{\varepsilon }*\cdots *g_{\varepsilon },}

where number of convolutions t/ε. result easy evaluate taking fourier transform of both sides, convolutions become multiplications:

k
~



(
p
;
t
)
=




g
~




ε


(
p

)

t

/

ε


.


{\displaystyle {\tilde {k}}(p;t)={\tilde {g}}_{\varepsilon }(p)^{t/\varepsilon }.}

the fourier transform of gaussian g gaussian of reciprocal variance:

g
~




ε


(
p
)
=

e

−



ε

p

2



2




,


{\displaystyle {\tilde {g}}_{\varepsilon }(p)=e^{-{\frac {\varepsilon p^{2}}{2}}},}

and result is

k
~



(
p
;
t
)
=

e

−



t

p

2



2




.


{\displaystyle {\tilde {k}}(p;t)=e^{-{\frac {tp^{2}}{2}}}.}

the fourier transform gives k, , gaussian again reciprocal variance:

k
(
x
−
y
;
t
)
∝

e

−



(
x
−
y

)

2




2
t





.


{\displaystyle k(x-y;t)\propto e^{-{\frac {(x-y)^{2}}{2t}}}.}

the proportionality constant not determined time-slicing approach, ratio of values different endpoint choices determined. proportionality constant should chosen ensure between each 2 time slices time evolution quantum-mechanically unitary, more illuminating way fix normalization consider path integral description of stochastic process.

the result has probability interpretation. sum on paths of exponential factor can seen sum on each path of probability of selecting path. probability product on each segment of probability of selecting segment, each segment probabilistically independently chosen. fact answer gaussian spreading linearly in time central limit theorem, can interpreted first historical evaluation of statistical path integral.

the probability interpretation gives natural normalization choice. path integral should defined that

∫
k
(
x
−
y
;
t
)

d
y
=
1.


{\displaystyle \int k(x-y;t)\,dy=1.}

this condition normalizes gaussian , produces kernel obeys diffusion equation:

d

d
t



k
(
x
;
t
)
=



∇

2


2


k
.


{\displaystyle {\frac {d}{dt}}k(x;t)={\frac {\nabla ^{2}}{2}}k.}

for oscillatory path integrals, ones in numerator, time slicing produces convolved gaussians, before. now, however, convolution product marginally singular, since requires careful limits evaluate oscillating integrals. make factors defined, easiest way add small imaginary part time increment ε. closely related wick rotation. same convolution argument before gives propagation kernel:

k
(
x
−
y
;
t
)
∝

e



i
(
x
−
y

)

2




2
t




,


{\displaystyle k(x-y;t)\propto e^{\frac {i(x-y)^{2}}{2t}},}

which, same normalization before (not sum-squares normalization – function has divergent norm), obeys free schrödinger equation:

d

d
t



k
(
x
;
t
)
=
i



∇

2


2


k
.


{\displaystyle {\frac {d}{dt}}k(x;t)=i{\frac {\nabla ^{2}}{2}}k.}

this means superposition of ks obey same equation, linearity. defining

ψ

t


(
y
)
=
∫

ψ

0


(
x
)
k
(
x
−
y
;
t
)

d
x
=
∫

ψ

0


(
x
)

∫

x
(
0
)
=
x


x
(
t
)
=
y



e

i
s



d
x
,


{\displaystyle \psi _{t}(y)=\int \psi _{0}(x)k(x-y;t)\,dx=\int \psi _{0}(x)\int _{x(0)=x}^{x(t)=y}e^{is}\,dx,}

then ψt obeys free schrödinger equation k does:

i


∂

∂
t




ψ

t


=
−



∇

2


2



ψ

t


.


{\displaystyle i{\frac {\partial }{\partial t}}\psi _{t}=-{\frac {\nabla ^{2}}{2}}\psi _{t}.}



simple harmonic oscillator

the lagrangian simple harmonic oscillator is

l


=



1
2



m




x
˙




2


−



1
2



m

ω

2



x

2


.


{\displaystyle {\mathcal {l}}={\tfrac {1}{2}}m{\dot {x}}^{2}-{\tfrac {1}{2}}m\omega ^{2}x^{2}.}

write trajectory x(t) classical trajectory plus perturbation, x(t) = xc(t) + δx(t) , action s = sc + δs. classical trajectory can written as

x

c


(
t
)
=

x

i





sin
⁡
ω
(

t

f


−
t
)


sin
⁡
ω
(

t

f


−

t

i


)



+

x

f





sin
⁡
ω
(
t
−

t

i


)


sin
⁡
ω
(

t

f


−

t

i


)



.


{\displaystyle x_{\text{c}}(t)=x_{i}{\frac {\sin \omega (t_{f}-t)}{\sin \omega (t_{f}-t_{i})}}+x_{f}{\frac {\sin \omega (t-t_{i})}{\sin \omega (t_{f}-t_{i})}}.}

this trajectory corresponds classical action:

s

c





=

∫


t

i





t

f






l



d
t
=

∫


t

i





t

f





(



1
2



m




x
˙




2


−



1
2



m

ω

2



x

2


)


d
t






=


1
2


m
ω

(



(

x

i


2


+

x

f


2


)
cos
⁡
ω
(

t

f


−

t

i


)
−
2

x

i



x

f




sin
⁡
ω
(

t

f


−

t

i


)



)

.






{\displaystyle {\begin{aligned}s_{\text{c}}&=\int _{t_{i}}^{t_{f}}{\mathcal {l}}\,dt=\int _{t_{i}}^{t_{f}}\left({\tfrac {1}{2}}m{\dot {x}}^{2}-{\tfrac {1}{2}}m\omega ^{2}x^{2}\right)\,dt\\[6pt]&={\frac {1}{2}}m\omega \left({\frac {(x_{i}^{2}+x_{f}^{2})\cos \omega (t_{f}-t_{i})-2x_{i}x_{f}}{\sin \omega (t_{f}-t_{i})}}\right).\end{aligned}}}

next, expand non-classical contribution action δs fourier series, gives

s
=

s

c


+

∑

n
=
1


∞





1
2




a

n


2




m
2



(



(
n
π

)

2





t

f


−

t

i





−

ω

2


(

t

f


−

t

i


)
)

.


{\displaystyle s=s_{\text{c}}+\sum _{n=1}^{\infty }{\tfrac {1}{2}}a_{n}^{2}{\frac {m}{2}}\left({\frac {(n\pi )^{2}}{t_{f}-t_{i}}}-\omega ^{2}(t_{f}-t_{i})\right).}

this means propagator is

k
(

x

f


,

t

f


;

x

i


,

t

i


)



=
q

e



i

s

c



ℏ




∏

j
=
1


∞





j
π


2



∫
d

a

j


exp
⁡


(


i

2
ℏ




a

j


2




m
2



(



(
j
π

)

2





t

f


−

t

i





−

ω

2


(

t

f


−

t

i


)
)

)








=

e



i

s

c



ℏ



q

∏

j
=
1


∞




(
1
−


(



ω
(

t

f


−

t

i


)


j
π



)


2


)


−


1
2










{\displaystyle {\begin{aligned}k(x_{f},t_{f};x_{i},t_{i})&=qe^{\frac {is_{\text{c}}}{\hbar }}\prod _{j=1}^{\infty }{\frac {j\pi }{\sqrt {2}}}\int da_{j}\exp {\left({\frac {i}{2\hbar }}a_{j}^{2}{\frac {m}{2}}\left({\frac {(j\pi )^{2}}{t_{f}-t_{i}}}-\omega ^{2}(t_{f}-t_{i})\right)\right)}\\[6pt]&=e^{\frac {is_{\text{c}}}{\hbar }}q\prod _{j=1}^{\infty }\left(1-\left({\frac {\omega (t_{f}-t_{i})}{j\pi }}\right)^{2}\right)^{-{\frac {1}{2}}}\end{aligned}}}

for normalization

q
=



m

2
π
i
ℏ
(

t

f


−

t

i


)




.


{\displaystyle q={\sqrt {\frac {m}{2\pi i\hbar (t_{f}-t_{i})}}}.}

using infinite product representation of sinc function:

∏

j
=
1


∞



(
1
−



x

2



j

2




)

=



sin
⁡
π
x


π
x



,


{\displaystyle \prod _{j=1}^{\infty }\left(1-{\frac {x^{2}}{j^{2}}}\right)={\frac {\sin \pi x}{\pi x}},}

the propagator can written as

k
(

x

f


,

t

f


;

x

i


,

t

i


)
=
q

e



i

s

c



ℏ







ω
(

t

f


−

t

i


)


sin
⁡
ω
(

t

f


−

t

i


)




=

e



i

s

c



ℏ







m
ω


2
π
i
ℏ
sin
⁡
ω
(

t

f


−

t

i


)




.


{\displaystyle k(x_{f},t_{f};x_{i},t_{i})=qe^{\frac {is_{\text{c}}}{\hbar }}{\sqrt {\frac {\omega (t_{f}-t_{i})}{\sin \omega (t_{f}-t_{i})}}}=e^{\frac {is_{c}}{\hbar }}{\sqrt {\frac {m\omega }{2\pi i\hbar \sin \omega (t_{f}-t_{i})}}}.}

let t = tf − ti. can write our propagator in terms of energy eigenstates as

k
(

x

f


,

t

f


;

x

i


,

t

i


)



=


(



m
ω


2
π
i
ℏ
sin
⁡
ω
t



)



1
2



exp
⁡


(


i
ℏ





1
2



m
ω



(

x

i


2


+

x

f


2


)
cos
⁡
ω
t
−
2

x

i



x

f




sin
⁡
ω
t



)








=

∑

n
=
0


∞


exp
⁡


(
−



i

e

n


t

ℏ


)



ψ

n


(

x

f



)

∗



ψ

n


(

x

i


)
.






{\displaystyle {\begin{aligned}k(x_{f},t_{f};x_{i},t_{i})&=\left({\frac {m\omega }{2\pi i\hbar \sin \omega t}}\right)^{\frac {1}{2}}\exp {\left({\frac {i}{\hbar }}{\tfrac {1}{2}}m\omega {\frac {(x_{i}^{2}+x_{f}^{2})\cos \omega t-2x_{i}x_{f}}{\sin \omega t}}\right)}\\[6pt]&=\sum _{n=0}^{\infty }\exp {\left(-{\frac {ie_{n}t}{\hbar }}\right)}\psi _{n}(x_{f})^{*}\psi _{n}(x_{i}).\end{aligned}}}

using identities sin ωt = 1/2e (1 − e) , cos ωt = 1/2e (1 + e),

k
(

x

f


,

t

f


;

x

i


,

t

i


)
=


(



m
ω


π
ℏ



)



1
2




e



−
i
ω
t

2





(
1
−

e

−
2
i
ω
t


)


−


1
2




exp
⁡


(
−



m
ω


2
ℏ




(

(

x

i


2


+

x

f


2


)




1
+

e

−
2
i
ω
t




1
−

e

−
2
i
ω
t





−



4

x

i



x

f



e

−
i
ω
t




1
−

e

−
2
i
ω
t





)

)


.


{\displaystyle k(x_{f},t_{f};x_{i},t_{i})=\left({\frac {m\omega }{\pi \hbar }}\right)^{\frac {1}{2}}e^{\frac {-i\omega t}{2}}\left(1-e^{-2i\omega t}\right)^{-{\frac {1}{2}}}\exp {\left(-{\frac {m\omega }{2\hbar }}\left(\left(x_{i}^{2}+x_{f}^{2}\right){\frac {1+e^{-2i\omega t}}{1-e^{-2i\omega t}}}-{\frac {4x_{i}x_{f}e^{-i\omega t}}{1-e^{-2i\omega t}}}\right)\right)}.}

we can absorb terms after first e r(t), thereby giving

k
(

x

f


,

t

f


;

x

i


,

t

i


)
=


(



m
ω


π
ℏ



)



1
2




e



−
i
ω
t

2



⋅
r
(
t
)
.


{\displaystyle k(x_{f},t_{f};x_{i},t_{i})=\left({\frac {m\omega }{\pi \hbar }}\right)^{\frac {1}{2}}e^{\frac {-i\omega t}{2}}\cdot r(t).}

we can expand r(t) in powers of e. terms in expansion multiplied e factor in front, , terms like

e



−
i
ω
t

2




e

−
i
n
ω
t


=

e

−
i
ω
t

(


1
2


+
n
)





for 

n
=
0
,
1
,
2
,
…
.


{\displaystyle e^{\frac {-i\omega t}{2}}e^{-in\omega t}=e^{-i\omega t\left({\frac {1}{2}}+n\right)}\quad {\text{for }}n=0,1,2,\ldots .}

comparing eigenstate expansion, energy spectrum simple harmonic oscillator:

e

n


=

(
n
+



1
2



)

ℏ
ω
.


{\displaystyle e_{n}=\left(n+{\tfrac {1}{2}}\right)\hbar \omega .}




coulomb potential

feynman s time-sliced approximation not, however, exist important quantum-mechanical path integrals of atoms, due singularity of coulomb potential e/r @ origin. after replacing time t path-dependent pseudo-time parameter

s
=
∫



d
t


r
(
t
)





{\displaystyle s=\int {\frac {dt}{r(t)}}}

the singularity removed , time-sliced approximation exists, integrable, since can made harmonic simple coordinate transformation, discovered in 1979 İsmail hakkı duru , hagen kleinert. combination of path-dependent time transformation , coordinate transformation important tool solve many path integrals , called generically duru–kleinert transformation.

the schrödinger equation

the path integral reproduces schrödinger equation initial , final state when potential present. easiest see taking path-integral on infinitesimally separated times.

ψ
(
y
;
t
+
ε
)
=

∫

−
∞


∞


ψ
(
x
;
t
)

∫

x
(
t
)
=
x


x
(
t
+
ε
)
=
y



e

i

∫

t


t
+
ε



(






x
˙




2


2


−
v
(
x
)
)


d
t



d
x
(
t
)

d
x

(
1
)


{\displaystyle \psi (y;t+\varepsilon )=\int _{-\infty }^{\infty }\psi (x;t)\int _{x(t)=x}^{x(t+\varepsilon )=y}e^{i\int \limits _{t}^{t+\varepsilon }\left({\frac {{\dot {x}}^{2}}{2}}-v(x)\right)\,dt}\,dx(t)\,dx\qquad (1)}

since time separation infinitesimal , cancelling oscillations become severe large values of ẋ, path integral has weight y close x. in case, lowest order potential energy constant, , kinetic energy contribution nontrivial. (this separation of kinetic , potential energy terms in exponent trotter product formula.) exponential of action is

e

−
i
ε
v
(
x
)



e

i






x
˙




2


2


ε




{\displaystyle e^{-i\varepsilon v(x)}e^{i{\frac {{\dot {x}}^{2}}{2}}\varepsilon }}

the first term rotates phase of ψ(x) locally amount proportional potential energy. second term free particle propagator, corresponding times diffusion process. lowest order in ε additive; in case 1 has (1):

ψ
(
y
;
t
+
ε
)
≈
∫
ψ
(
x
;
t
)

e

−
i
ε
v
(
x
)



e



i
(
x
−
y

)

2




2
ε





d
x

.


{\displaystyle \psi (y;t+\varepsilon )\approx \int \psi (x;t)e^{-i\varepsilon v(x)}e^{\frac {i(x-y)^{2}}{2\varepsilon }}\,dx\,.}

as mentioned, spread in ψ diffusive free particle propagation, infinitesimal rotation in phase varies point point potential:

∂
ψ


∂
t



=
i
⋅

(



1
2




∇

2


−
v
(
x
)
)

ψ



{\displaystyle {\frac {\partial \psi }{\partial t}}=i\cdot \left({\tfrac {1}{2}}\nabla ^{2}-v(x)\right)\psi \,}

and schrödinger equation. note normalization of path integral needs fixed in same way in free particle case. arbitrary continuous potential not affect normalization, although singular potentials require careful treatment.

equations of motion

since states obey schrödinger equation, path integral must reproduce heisenberg equations of motion averages of x , ẋ variables, instructive see directly. direct approach shows expectation values calculated path integral reproduce usual ones of quantum mechanics.

start considering path integral fixed initial state

∫

ψ

0


(
x
)

∫

x
(
0
)
=
x



e

i
s
(
x
,



x
˙



)



d
x



{\displaystyle \int \psi _{0}(x)\int _{x(0)=x}e^{is(x,{\dot {x}})}\,dx\,}

now note x(t) @ each separate time separate integration variable. legitimate change variables in integral shifting: x(t) = u(t) + ε(t) ε(t) different shift @ each time ε(0) = ε(t) = 0, since endpoints not integrated:

∫

ψ

0


(
x
)

∫

u
(
0
)
=
x



e

i
s
(
u
+
ε
,



u
˙



+



ε
˙



)



d
u



{\displaystyle \int \psi _{0}(x)\int _{u(0)=x}e^{is(u+\varepsilon ,{\dot {u}}+{\dot {\varepsilon }})}\,du\,}

the change in integral shift is, first infinitesimal order in ε:

∫

ψ

0


(
x
)

∫

u
(
0
)
=
x



(
∫



∂
s


∂
u



ε
+



∂
s


∂



u
˙









ε
˙




d
t
)


e

i
s



d
u



{\displaystyle \int \psi _{0}(x)\int _{u(0)=x}\left(\int {\frac {\partial s}{\partial u}}\varepsilon +{\frac {\partial s}{\partial {\dot {u}}}}{\dot {\varepsilon }}\,dt\right)e^{is}\,du\,}

which, integrating parts in t, gives:

∫

ψ

0


(
x
)

∫

u
(
0
)
=
x


−

(
∫

(


d

d
t






∂
s


∂



u
˙






−



∂
s


∂
u



)

ε
(
t
)

d
t
)


e

i
s



d
u



{\displaystyle \int \psi _{0}(x)\int _{u(0)=x}-\left(\int \left({\frac {d}{dt}}{\frac {\partial s}{\partial {\dot {u}}}}-{\frac {\partial s}{\partial u}}\right)\varepsilon (t)\,dt\right)e^{is}\,du\,}

but shift of integration variables, doesn t change value of integral choice of ε(t). conclusion first order variation 0 arbitrary initial state , @ arbitrary point in time:

⟨

ψ

0



|



δ
s


δ
x



(
t
)
|


ψ

0


⟩

=
0


{\displaystyle \left\langle \psi _{0}\left|{\frac {\delta s}{\delta x}}(t)\right|\psi _{0}\right\rangle =0}

this heisenberg equation of motion.

if action contains terms multiply ẋ , x, @ same moment in time, manipulations above heuristic, because multiplication rules these quantities noncommuting in path integral in operator formalism.

stationary-phase approximation

if variation in action exceeds ħ many orders of magnitude, typically have destructive interference other in vicinity of trajectories satisfying euler–lagrange equation, reinterpreted condition constructive interference. can shown using method of stationary phase applied propagator. ħ decreases, exponential in integral oscillates rapidly in complex domain change in action. thus, in limit ħ goes zero, points classical action not vary contribute propagator.

canonical commutation relations

the formulation of path integral not make clear @ first sight quantities x , p not commute. in path integral, these integration variables , have no obvious ordering. feynman discovered non-commutativity still present.

to see this, consider simplest path integral, brownian walk. not yet quantum mechanics, in path-integral action not multiplied i:

s
=
∫


(



d
x


d
t



)


2



d
t


{\displaystyle s=\int \left({\frac {dx}{dt}}\right)^{2}\,dt}

the quantity x(t) fluctuating, , derivative defined limit of discrete difference.

d
x


d
t



=



x
(
t
+
ε
)
−
x
(
t
)

ε




{\displaystyle {\frac {dx}{dt}}={\frac {x(t+\varepsilon )-x(t)}{\varepsilon }}}

note distance random walk moves proportional √t, that:

x
(
t
+
ε
)
−
x
(
t
)
≈


ε




{\displaystyle x(t+\varepsilon )-x(t)\approx {\sqrt {\varepsilon }}}

this shows random walk not differentiable, since ratio defines derivative diverges probability one.

the quantity xẋ ambiguous, 2 possible meanings:

[
1
]
=
x



d
x


d
t



=
x
(
t
)



x
(
t
+
ε
)
−
x
(
t
)

ε




{\displaystyle [1]=x{\frac {dx}{dt}}=x(t){\frac {x(t+\varepsilon )-x(t)}{\varepsilon }}}








[
2
]
=
x



d
x


d
t



=
x
(
t
+
ε
)



x
(
t
+
ε
)
−
x
(
t
)

ε




{\displaystyle [2]=x{\frac {dx}{dt}}=x(t+\varepsilon ){\frac {x(t+\varepsilon )-x(t)}{\varepsilon }}}

in elementary calculus, 2 different amount goes 0 ε goes 0. in case, difference between 2 not 0:

[
2
]
−
[
1
]
=





(


x
(
t
+
ε
)
−
x
(
t
)



)



2



ε


≈


ε
ε




{\displaystyle [2]-[1]={\frac {{\big (}x(t+\varepsilon )-x(t){\big )}^{2}}{\varepsilon }}\approx {\frac {\varepsilon }{\varepsilon }}}

give name value of difference 1 random walk:

(


x
(
t
+
ε
)
−
x
(
t
)



)



2



ε


=
f
(
t
)


{\displaystyle {\frac {{\big (}x(t+\varepsilon )-x(t){\big )}^{2}}{\varepsilon }}=f(t)}

and note f(t) rapidly fluctuating statistical quantity, average value 1, i.e. normalized gaussian process . fluctuations of such quantity can described statistical lagrangian

l


=
(
f
(
t
)
−
1

)

2



,


{\displaystyle {\mathcal {l}}=(f(t)-1)^{2}\,,}

and equations of motion f derived extremizing action s corresponding l set equal 1. in physics, such quantity equal 1 operator identity . in mathematics, weakly converges 1 . in either case, 1 in expectation value, or when averaged on interval, or practical purpose.

defining time order operator order:

[
x
,



x
˙



]
=
x



d
x


d
t



−



d
x


d
t



x
=
1


{\displaystyle [x,{\dot {x}}]=x{\frac {dx}{dt}}-{\frac {dx}{dt}}x=1}

this called itō lemma in stochastic calculus, , (euclideanized) canonical commutation relations in physics.

for general statistical action, similar argument shows that

[
x
,



∂
s


∂



x
˙






]

=
1


{\displaystyle \left[x,{\frac {\partial s}{\partial {\dot {x}}}}\right]=1}

and in quantum mechanics, imaginary unit in action converts canonical commutation relation,

[
x
,
p
]
=
i


{\displaystyle [x,p]=i}



particle in curved space

for particle in curved space kinetic term depends on position, , above time slicing cannot applied, being manifestation of notorious operator ordering problem in schrödinger quantum mechanics. 1 may, however, solve problem transforming time-sliced flat-space path integral curved space using multivalued coordinate transformation (nonholonomic mapping explained here).

measure-theoretic factors

sometimes (e.g. particle moving in curved space) have measure-theoretic factors in functional integral:

∫
μ
[
x
]

e

i
s
[
x
]





d


x
.


{\displaystyle \int \mu [x]e^{is[x]}\,{\mathcal {d}}x.}

this factor needed restore unitarity.

for instance, if

s
=
∫

(


m
2



g

i
j






x
˙




i






x
˙




j


−
v
(
x
)
)


d
t
,


{\displaystyle s=\int \left({\frac {m}{2}}g_{ij}{\dot {x}}^{i}{\dot {x}}^{j}-v(x)\right)\,dt,}

then means each spatial slice multiplied measure √g. measure cannot expressed functional multiplying dx measure because belong entirely different classes.

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