1

Completing Canonical Quantization

John R. KlauderUniversity of Florida

Gainesville, FL

2

Something Strange

L. Landau, E.M. Lifshitz, Quantum mechanics: Non-relativistic theory, 3rd ed., Pergamon Press, 1977.

"Thus quantum mechanics occupies a very unusual place among physical theories: it contains classical mechanics as a limiting case, yet at the same time it requires this limiting case for its own formulation."

3

Classical & Quantum

0classical

quantum

0

4

DIRAC

The Principles of Quantum Mechanics

5

Classical Quantum

classical

quantum

0

6

List of Topics

1 Classical/Quantum connection

“Enhanced Quantization”

Canonical & Affine quantization Enhanced classical theories

2 Two toy models

3 Rotationally symmetric models

7

TOPIC 1

• Classical & Quantum formalism• Canonical coherent states• Classical Quantum formalism• Canonical transformations• Cartesian coordinates• Affine vs. canonical variables• Affine quantization as canonical

quantization

8

Action Principle Formulations

H

R

(0)given )( :Solution

/ yields 0 :Variation

})(]/[)({ :action Quantum

)0( ),0(given )( ),( :Solution

/ ,/ :yields 0 :Variation

))](),(()()([ :action Classical

2

t

tiA

dtttitA

qptqtp

qHppHqA

dttqtpHtqtpA

Q

Q

ccC

cC

H

H

VERY DIFFERENT

9

Restricted Action Principle

H

H

H

EE

E

E

E

EE

E

EEE

E

subsett

subspacet

t

tttiA

ttt

dtttitA

Q

Q

:})({ of Nature (2)

:})({ of Nature (1)

(0)given )( :Solution

] [

])( /)([ yields 0 :Variation

])( [ )()( :nsrestrictio Possible

})(]/[)({ :action Quantum

HH

H

H

(Gaussians)

(half space)

10

Unification of Classical and Quantum (1)

HS

)( )( : variationRestricted

)(]/[)( :action Quantum

tt

dtttitAQ

H

)(xx )(; qxqx

?

Macroscopic variations of Microscopic states:

Basic state: Translated basic state: Translated Fourier state: Coherent states:

)(~; pkpk )(, /)( qxeqpx qxip

00)( ; 0 ; , // iPQeeqp ipQiqP s.a.

11

Unification of Classical and Quantum (2)

dttqtpHtqtpA

dttqtptitqtpA

tqtpt

dtttitA

R

R

Q

))](),(()()([

)(),(]/[)(),( :action New

)(),()( : variationRestricted

)(]/[)( :action Quantum

H

H

CLASSICAL MECHANICS IS QUANTUM MECHANICS RESTRICTED TO A CERTAIN TWO DIMENSIONAL SURFACE IN HILBERT SPACE

subset

12

Canonical Transformations

dtqpHqpGqp

dtqpQPtiqpA

qpqpqqppqp

qpGdqdpdqp

dtqpHqp

dtqpQPtiqpA

R

R

)]~,~(~)~,~(~~~[

~,~)],(/[~,~

:action quantum Restricted

~,~)~,~(),~,~( ,

)~,~(~~~ :ations transformCanonical

)],([

,)],(/[,

:action quantum Restricted

H

H

13

Cartesian Coordinates

!ntization onical quational cant to tradi Equivalen

dqdpqpdqpqpd

eD

qqpQqppqpPqp

QP

qpqpqQpP

qpQPqpqpH

iR

]|,,|||,||[2

] min[ :metricStudy -Fubini

,, ; ,,

] 000 , 000 [ :meaning Physical

),;(),(0),(0

,),(,),(

:connection QuantumClassical/

2222

22

OHH

H

14

Quantum/Classical Summary

qqpHppqpHq

dttqtpHtqtp

dttqtpQPtitqtpA

eetqtpt

tQPtti

dttQPtitA

restQ

QtipPtiq

Q

/),( , /),(

))](),(()()([

)(),()],(/[)(),(

0)(),()(

:action quantum Restricted

)(),(/)(

)()],(/[)(

:action Qauntum

.

/)(/)(

H

H

H

15

Is There More?

• Are there other two-dimensional sheets of normalized Hilbert space vectors that may be used in restricting the quantum action and which lead to an enhanced classical canonical formalism?

16

Is There More?

• Are there other two-dimensional sheets of normalized Hilbert space vectors that may be used in restricting the quantum action and which lead to an enhanced classical canonical formalism?

YES !

17

Affine Variables

)/~

2/11/(2/,,

:unity of Resolution

]}~

/)'(''//'[{,','

:function Overlap

0]~/)1[(; ,

)0 ; 0( :statescoherent Affine

2/)( ; ],[],[],[

},{},{},{ : variablesAffine

/~221

/)ln(/

dqdpqpqpI

ppqqiqqqqqpqp

iDQeeqp

Qq

QPPQDDQQPQPQQQi

dqpqqpqqq

DqiipQ

[(

also (q < 0 , Q < 0) U (q > 0 , Q > 0)

s.a.

18

Affine Quantization (1)

))](~),(~(~))(~),(~('~)(~)(~[

)(~),(~)],(/[)(~),(~

,~,~ :ation transformCanonical

))](),(()()([

)(),()],(/[)(),(

)(),()( :action Restricted

)()],(/[)(

:action Quantum

dttqtpHtqtpGtptq

dttqtpQDtitqtpA

qpqp

dttqtpHtptq

dttqtpQDtitqtpA

tqtpt

dttQDtitA

R

R

Q

H'

H'

H'

subset

19

Affine Quantization (2)

2

2222122

/~

, :Cartesian becomes Metric

~~]|, ,|||,||[2

:metricStudy -Fubini

,, ; ,,

] 1 , 0 [ :meaning Physical

),;(),(),(

,),(,),('

:connection QuantumClassical/

qq

dqqdpqqpdqpqpd

qqpQqppqqpDqp

QD

qpqpqqQpqQD

qpQDqpqpqH

OH'H'

H'

20

The Q/C Connection : Summary

• The classical action arises by a restriction of the quantum action to coherent states

• Canonical quantization uses P and Q which must be self adjoint

• Affine quantization uses D and Q which are self adjoint when Q > 0 (and/or Q < 0)

• Both canonical AND affine quantum versions are consistent with classical, canonical phase space variables p and q

• Now for some applications!

21

TOPIC 2

• Solutions of the first model have singularities

• Canonical quantum corrections• Affine quantum corrections• Affine quantization resolves singularities!• A second classical model is similar

200

100

2

02

)1()( , )1()(

)()(

0)( , ])()()()([

tpqtqtpptp

tptp

tqdttptqtptqA TC

22

Toy Model - 1

qQQeeqQQee

EMECKDDQC

KtMtqKt

ttp

qCqpqpDDQqp

taaEtqtaatp

aqaqpqpPQPqp

tpqtqtpptp

qdtqppqA

DqiDqiiqPiqP

C

/)ln(/)ln(//

020

212

22

221

220

222

200

100

2

;

4 ; 4/ ;

0])[()( , )(

)()( :Solution

/,, :quant. Affine

))(sin()/()( , ))(cot()( :Solution

2/ ; ,, :quant. Canonical

)1()( , )1()( :Solution

0 ; ][ :action. Classical

23

Toy Model - 2

)( )/('

' ; |Q|/

/'||/,||/, :Model

)||,||/(,),(,),(

)],([,)],(/[,

])( ; 0 [ :action classical Extended

][ ; |)]|/([ :modelToy

0|| ,0|| ; , :statescoherent Affine

)( , ],[],[ :onquantizati Affine

22

2122122

222122

21

2221

/|)ln(|/

21

CsBohr radiuCmeC

PCeeC

qCqCpqpQePqp

QqpqPqpQPqpqpH

dtqpHpqdtqpQPtiqpA

abDaQ

dtqeppqA

qQeeqp

PQQPDDQQiQPQ

mm

mm

R

mC

DqiipQ

HH

H

24

Enhanced Toy Models : Summary

• Classical toy models exhibit singular solutions for all positive energies

• Enhanced classical theory with canonical quantum corrections still exhibits singularities

• Enhanced classical theory with affine quantum corrections removes all singularities

• Enhanced quantization can eliminate singularities

25

TOPIC 3

• Rotationally symmetric models

• Free quantum models for • Interacting quantum models for

• Reducible operator representation is the key

,) ,,(,),(

,,),(

}{ , )(][),( 122

022

02

21

qpQPqpqpH

qpqpqpH

N

N

ppqqmpqpH Nnn

H

H

][ 2 ppp

. .

.

26

Rotationally Sym. Models (1)

NthatnecessaryisitNWhen

NQQmP

iPQQqPp

YXZqpLE

qZqpYpX

RNOqOqpOp

NqqmpqpH

qqqppp

jkkj

NN

/,

,:)(::::nHamiltonia

],[;, :onQuantizati

)(, :motion of Constants

,, :invariants Basic

),(;,under Invariant

;)(][),(

),...,(,),...,( :scoordinate space Phase

0

220

220

221

222

22

220

220

221

11

H

O

27

Rotationally Sym. Models (2)

; )( :Result

m)2/1()( :Uniqueness

]1)([ ; )(

)(

)sin()(

)()(

ondistributi state ground theofon ansformatiFourier tr

).()( : with )(

real aith equation wer Schroeding

m4/

002/

212)2(2/

2212/)cos(

2/

2

2

222

p

bp

NN

NNrp

N

NNN

Nipr

Nxpi

N

NNN

epC

bbf

dbbfdbbfe

ddrrreK

dddrrre

xdxepC

xr symmetryrotationalfullx

und stateunique gro

A free theory!

28

…Now, Do Some Hard Work…

29

Rotationally Sym. Models (3)

Nqqmp

qmvqmqmp

qpQSmRvQSmR

SQmPqpqpqp

PqQpiqp

RiQSmPiSQm

Nqmpwqmp

qpQmPwQmPqpqpqp

PqQpiqpPiQm

;)()(

)()(

;,:}])([::)(:

:)(:{;,;,;,

;0]/)(exp[;,

10;0;0])([;0])([

;)()(

,:})(:::{,,~,

0]/)(exp[, ; 00)(

220

220

221

224422221222

21

222222221

22221

2220

2220

221

2220

2220

221

0

H'

HTEST

REAL

30

Rot. Sym. Models : Summary• Conventional quantization works if N is

finite but leads to triviality if N is infinite• Enhanced quantization applies even for

reducible operator representations• Using the Weak Correspondence

Principle a nontrivial quantization results if N is finite

or N is infinite --- with NO divergences !• Class. & Quant. formalism is similar for all N

qpqpqpH

,,),( H

WHAT HAS BEEN ACCOMPLISHED ??

31

• Canonical quantization requires Cartesian coordinates, but WHY is not clear

• Canonical quantization works well for certain problems, but NOT for all problems

• Enhanced quantization clarifies coordinate transformations and Cartesian coordinates

• Enhanced quantization can yield canonical results -- OR provide proper results when canonical quantization fails

Canonical vs. Enhanced

32

Other Enh. Quant. Projects

• Simple models of affine quantization eliminating classical singularities (on going)

• Covariant scalar models (done)• Affine quantum gravity (started)• Incorporating constrained systems within

enhanced quantization (started)• Additional sheets of vectors in Hilbert space

relating quan. and class. models (started)• Extension to fermion fields (hints)

4n

33

Main Message of Today

c

q q

c

34

Thank You

35

References

• ``Enhanced Quantization: A Primer'', J. Phys. A: Math. Theor. 45, 285304

(8pp) (2012); arXiv:1204.2870• ``Enhanced Quantization on the Circle’’; Phys.

Scr. 87 035006 (5pp) (2013); arXiv:1206.1180• ``Enhanced Quantum Procedures that

Resolve Difficult Problems’’; arXiv: 1206.4017• ``Revisiting Canonical Quantization’’; arXiv:

1211.735