the path integral for relativistic worldlines -...

The Path Integral for

Relativistic WorldlinesB. Koch

with E. Muñoz and I. Reyes based on:

Phys.Rev. D96 (2017) no.8, 085011 and arXiv:1706.05388.

1

Afunalhue, La parte y el todo, 2018

Content

PI of the RPP, Status

Local Symmetry: Velocity Rotations

Constructing the PI of the RPP

Conclusion

2

The Path Integral

A B

Propagator

hA|Bi ⇠Z

Dx e

�SA,B

(after Wick rotation)3

Non Relativistic Propagator

S =

Z tB ,B

tA,Adt

m

2(~̇x)2

hA|Bi ⇠Z

Dx e

�SA,B

Two Nice Features

Quadratic in field variable

Can be connected (Chapman, Kolmogorov)

=X

i

�m

2

(~xi+1 � ~xi)2

�2

4

Feature 1: Quadratic

Non Relativistic Propagator

hA|Bi = ⇧i

(Ni ·

Zd

dxi e

�✓P

i

�m

2

(~xi+1�~x

i

)2

�2

◆)

Simple Gaussian integrals

Feature 2: Chapman Kolmogorov

hA|Ci =Z

d

dxbhA|BihB|Ci

Probability conservation&

stepwise construction of PI5

Relativistic Propagator

hA|Bi ⇠Z

Dx e

�SA,B

Two Problems!

S =

Zd�

s✓dx

µ

d�

◆2

6

Why intersting, why here

Unsolved NON PERTURBATIVE problem

Simplest system with general covariance

Problem 1: Square root


Horrible integrals & still wrong result

Problem 2: No Chapman Kolmogorov

No probability conservation&

no stepwise construction of PI

hA|Bi =? ⇧i

8><

>:Ni ·

Zd

dxi e

� P

i

�

r(x

µ

i+1�x

µ

i

)2

�2

!9>=

>;

hA|Ci 6=Z

d

dxBhA|BihB|Ci

7


„Solutions“ in the Literature

Hamiltonian formalism (classically equivalent) Evades P1

Solves P1 & P2, but high price

Interesting, neither P1 nor P2 are solved

Ignore the problems and do QFT right away

Thats what we mostly do …

*1

Redefine probability *2

Restrict PI to spheres, or other approx.*3

8

Relativistic PI: our Proposal

A B

hA|Bi ⇠Z

Dx e

�SA,B

9


hA|Bi ⇠Z

Dx e

�SA,B

S =

Zd�

s✓dx

µ

d�

◆2

Issue 1: Local reparametrizations (known)

Issue 2: Local velocity rotations (trivial?)

Issue 3: Measure without anomalies10

with

Three issues:

can be done, if one considers


hA|Bi ⇠Z

Dx e

�SA,B

S =

Zd�

s✓dx

µ

d�

◆2

11

using I1,I2,I3 works

Functional Fadeev Popov method

*0 Geometricstepwise proof

*00

S =

Zd�

s✓dx

µ

d�

◆2

13

Issue 1: Local reparametrizations (known)

Invariant under � ! �0(�)

We fix proper time such that

⌧ = ⌧(�)✓dx

µ

d⌧

◆2

= 1with

Geometric over counting:

A B A BC

14

Issue 2: Local velocity rotations (trivial?)

S =

Zd�

s✓dx

µ

d�

◆2

=

Zd�

pv

µvµ

Invariant under vµ ! v0µ = ⇤µ⌫(�)v

⌫

v0µv0µ = vµvµwith

Factor out of PI

. . .. . .

. . .. . .

if and ! S = S0 L = L0

15

Issue 3: Measure without anomalies

When performing transformation

vµ ! v0µ = ⇤µ⌫(�)v

⌫

define right measure invariant under this symmetry:

Dx ! Dx

0 = Dx

Geometric example for two step propagator

16

Stepwise proof

A BhA|Bi

=

= hA|Bi1

+ + . . .

hA|Bi2+ + . . .

Strategy


17

hA|Bi1

~x1

A B

Change of integration coordinates

(x1, y1) ! (S,↵)

x1 =

S

2M

cos(↵)

y1 =xf

2

s✓S

xfM

◆2

� 1 · sin(↵)

xf = | ~B � ~

A|

= N

Zd

2x1 �1 e

�(SA,~x1

+S~x1,B

)

Calculate using I2,3 hA|Bi1

18

~x1

A B

Change of integration coordinates

(x1, y1) ! (S,↵)

x1 =

S

2M

cos(↵)

y1 =xf

2

s✓S

xfM

◆2

� 1 · sin(↵)

xf = | ~B � ~

A|

h0, ~xf

i1 = N2,1(ti,f )

Z 1

xfM

dS

Z 2⇡

0d↵ ·�1

2 (S/M)

2 � x

2f

(1 + cos(2↵))

8

p(S)

2 � (x

f

M)

2exp [�S]

↵S


~x1

h0, ~xf

i1 = N2,1(ti,f )

Z 1

xfM

dS

Z 2⇡

0d↵ ·�1

2 (S/M)

2 � x

2f

(1 + cos(2↵))

8

p(S)

2 � (x

f

M)

2exp [�S]

↵S

0 xf

↵ ! ↵0For one sees and L = L0S = S0

I2 velocity rotation!Naive, measure depends on : anomaly I3 ↵

cancels anomaly��11 ⌘ |~x1 � 0| · |~xf � ~x1|


19

↵S

0 xf

↵ ! ↵0For nothing changes

h0, ~xf

i1 = N2,1

✓Z 2⇡

0d↵

◆·Z 1

xfM

dS

1p(S)

2 � (x

f

M)

2exp [�S]

h0, ~xf i1 = N ·K0(xfM)


20

S

0 xf


h0, ~xf

i1 = N2,1

✓Z 2⇡

0d↵

◆·Z 1

xfM

dS

1p(S)

2 � (x

f

M)

2exp [�S]

h0, ~xf i1 = N ·K0(xfM)

↵0


21

S

0 xf


h0, ~xf

i1 = N2,1

✓Z 2⇡

0d↵

◆·Z 1

xfM

dS

1p(S)

2 � (x

f

M)

2exp [�S]

h0, ~xf i1 = N ·K0(xfM)

↵0


22

23

Stepwise proof

A BhA|Bi

=

= hA|Bi1

+ + . . .

hA|Bi2+ + . . .

Strategy


25


0xfh0|xf i2 :

~x1

~x2

↵

S1,f

↵ ! ↵0For nothing changes ~x2 ! ~x

02

26


0xf

~x1

h0|xf i2 :S1,f

↵ ! ↵0For nothing changes (I2) ~x2 ! ~x

02

~x

02

↵0

straight line: I1 reparametrizations0 ! ~x1 ! ~x

02

27


0xfh0|xf i2 :

~x1

S1,f

↵ ! ↵0For nothing changes (I2) ~x2 ! ~x

02

~x

02

straight line: I1 reparametrizations0 ! ~x1 ! ~x

02

h0|xf i2 ⇢1,2 h0|xf i1

Concluding Comments

Generalization to D dimensions

PI of RPP action can be done, considering I1,I2,I3

Chapman Kolmogorov becomes „trivial“

Future work …

29

S

0 xf↵0

Thank You

30

Literature

31

0) B. K., E. Muñoz and I. Reyes; Phys.Rev. D96 (2017) no.8, 08501100) B. K., E. Muñoz; arXiv:1706.05388.1) J. Polchinski, “String Theory”, Cambridge U. P., ISBN 0521-63303-6, page 145.; H. Kleinert, “Path Integrals in Quantum Mechanics…”, World Scientific Publishing, ISBN 978-981-4273-55-8, page 1359–1369. M. Henneaux and C. Teitelboim, Annals Phys. 143, 127 (1982).2) P. Jizba and H. Kleinert, Phys. Rev. E 78, 031122 (2008). 3) E. Prugovecki, Il Nuovo Cimento, 61 A, N.2, 85 (1981). H. Fukutaka and T. Kashiwa, Annals of Physics, 176, 301 (1987). Padmanabhan, T. Found Phys (1994) 24: 1543.

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