1. The document discusses the unification of classical and quantum mechanics through restricting the quantum action to certain two-dimensional surfaces in Hilbert space, such as coherent states. 2. Both canonical and affine quantization are considered. Affine quantization uses affine variables like D and Q which are self-adjoint for certain domains, resolving singularities. 3. Two toy models are examined to illustrate how affine quantum corrections can remove singularities compared to canonical corrections. The affine quantization approach provides a unified description of classical and quantum mechanics.

1. 1
Completing
Canonical Quantization
John R. Klauder
University of Florida
Gainesville, FL

2. 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. 3
Classical & Quantum
0


classical
quantum
0



4. 4
DIRAC
The
Principles of
Quantum
Mechanics

5. 5
Classical Quantum

classical
quantum
0



6. 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. 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. 8
Action Principle Formulations
H
R











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
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
(0)
given
)
(
:
Solution
/
yields
0
:
Variation
}
)
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]
/
[
)
(
{
:
action
Quantum
)
0
(
),
0
(
given
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),
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:
Solution
/
,
/
:
yields
0
:
Variation
))]
(
),
(
(
)
(
)
(
[
:
action
Classical
2








t
t
i
A
dt
t
t
i
t
A
q
p
t
q
t
p
q
H
p
p
H
q
A
dt
t
q
t
p
H
t
q
t
p
A
Q
Q
c
c
C
c
C
H
H





VERY DIFFERENT

9. 9
Restricted Action Principle
H
H
H
E
E
E
E
E
E
E
E
E
E
E
E




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
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

subset
t
subspace
t
t
t
t
t
i
A
t
t
t
dt
t
t
i
t
A
Q
Q
:
}
)
(
{
of
Nature
(2)
:
}
)
(
{
of
Nature
(1)
(0)
given
)
(
:
Solution
]
[
]
)
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/
)
(
[
yields
0
:
Variation
]
)
(
[
)
(
)
(
:
ns
restrictio
Possible
}
)
(
]
/
[
)
(
{
:
action
Quantum












H
H
H
H


(Gaussians)
(half space)

10. 10
Unification of
Classical and Quantum (1)
H
S 







)
(
)
(
:
variation
Restricted
)
(
]
/
[
)
(
:
action
Quantum
t
t
dt
t
t
i
t
AQ


 H

)
(x
x 
 
)
(
; q
x
q
x 


?
Macroscopic variations of Microscopic states:
Basic state:
Translated basic state:
Translated Fourier state:
Coherent states:
)
(
~
; p
k
p
k 


)
(
, /
)
(
q
x
e
q
p
x q
x
ip

 


0
0
)
(
;
0
;
, /
/



 
iP
Q
e
e
q
p ipQ
iqP



 s.a.

11. 11
Unification of
Classical and Quantum (2)
dt
t
q
t
p
H
t
q
t
p
A
dt
t
q
t
p
t
i
t
q
t
p
A
t
q
t
p
t
dt
t
t
i
t
A
R
R
Q
 












))]
(
),
(
(
)
(
)
(
[
)
(
),
(
]
/
[
)
(
),
(
:
action
New
)
(
),
(
)
(
:
variation
Restricted
)
(
]
/
[
)
(
:
action
Quantum



H
H



CLASSICAL MECHANICS IS QUANTUM
MECHANICS RESTRICTED TO A CERTAIN TWO
DIMENSIONAL SURFACE IN HILBERT SPACE
subset

12. 12
Canonical Transformations
dt
q
p
H
q
p
G
q
p
dt
q
p
Q
P
t
i
q
p
A
q
p
q
p
q
q
p
p
q
p
q
p
G
d
q
d
p
dq
p
dt
q
p
H
q
p
dt
q
p
Q
P
t
i
q
p
A
R
R
 











 






)]
~
,
~
(
~
)
~
,
~
(
~
~
~
[
~
,
~
)]
,
(
/
[
~
,
~
:
action
quantum
Restricted
~
,
~
)
~
,
~
(
),
~
,
~
(
,
)
~
,
~
(
~
~
~
:
ations
transform
Canonical
)]
,
(
[
,
)]
,
(
/
[
,
:
action
quantum
Restricted





H
H

13. 13
Cartesian Coordinates
!
ntization
onical qua
tional can
t to tradi
Equivalen
dq
dp
q
p
d
q
p
q
p
d
e
D
q
q
p
Q
q
p
p
q
p
P
q
p
Q
P
q
p
q
p
q
Q
p
P
q
p
Q
P
q
p
q
p
H
i
R
]
|
,
,
|
||
,
||
[
2
]
min
[
:
metric
Study
-
Fubini
,
,
;
,
,
]
0
0
0
,
0
0
0
[
:
meaning
Physical
)
,
;
(
)
,
(
0
)
,
(
0
,
)
,
(
,
)
,
(
:
connection
Quantum
Classical/
2
2
2
2
2
2


















 

O
H
H
H

14. 14
Quantum/Classical Summary
q
q
p
H
p
p
q
p
H
q
dt
t
q
t
p
H
t
q
t
p
dt
t
q
t
p
Q
P
t
i
t
q
t
p
A
e
e
t
q
t
p
t
t
Q
P
t
t
i
dt
t
Q
P
t
i
t
A
rest
Q
Q
t
ip
P
t
iq
Q






 


















/
)
,
(
,
/
)
,
(
))]
(
),
(
(
)
(
)
(
[
)
(
),
(
)]
,
(
/
[
)
(
),
(
0
)
(
),
(
)
(
:
action
quantum
Restricted
)
(
)
,
(
/
)
(
)
(
)]
,
(
/
[
)
(
:
action
Qauntum
.
/
)
(
/
)
(








H
H
H






15. 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. 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. 17
Affine Variables
)
/
~
2
/
1
1
/(
2
/
,
,
:
unity
of
Resolution
]}
~
/
)
'
(
'
'
/
/
'
[
{
,
'
,
'
:
function
Overlap
0
]
~
/
)
1
[(
;
,
)
0
;
0
(
:
states
coherent
Affine
2
/
)
(
;
]
,
[
]
,
[
]
,
[
}
,
{
}
,
{
}
,
{
:
variables
Affine
/
~
2
2
1
/
)
ln(
/




































dq
dp
q
p
q
p
I
p
p
q
q
i
q
q
q
q
q
p
q
p
iD
Q
e
e
q
p
Q
q
QP
PQ
D
D
Q
QP
Q
P
Q
Q
Q
i
d
q
pq
q
p
q
q
q
D
q
i
ipQ
also (q < 0 , Q < 0) U (q > 0 , Q > 0)
s.a.

18. 18
Affine Quantization (1)
))]
(
~
),
(
~
(
~
))
(
~
),
(
~
(
'
~
)
(
~
)
(
~
[
)
(
~
),
(
~
)]
,
(
/
[
)
(
~
),
(
~
,
~
,
~
:
ation
transform
Canonical
))]
(
),
(
(
)
(
)
(
[
)
(
),
(
)]
,
(
/
[
)
(
),
(
)
(
),
(
)
(
:
action
Restricted
)
(
)]
,
(
/
[
)
(
:
action
Quantum
dt
t
q
t
p
H
t
q
t
p
G
t
p
t
q
dt
t
q
t
p
Q
D
t
i
t
q
t
p
A
q
p
q
p
dt
t
q
t
p
H
t
p
t
q
dt
t
q
t
p
Q
D
t
i
t
q
t
p
A
t
q
t
p
t
dt
t
Q
D
t
i
t
A
R
R
Q


 








 


















H'
H'
H'



subset

19. 19
Affine Quantization (2)



















2
2
2
2
2
1
2
2
/
~
,
:
Cartesian
becomes
Metric
~
~
]
|
,
,
|
||
,
||
[
2
:
metric
Study
-
Fubini
,
,
;
,
,
]
1
,
0
[
:
meaning
Physical
)
,
;
(
)
,
(
)
,
(
,
)
,
(
,
)
,
(
'
:
connection
Quantum
Classical/














q
q
dq
q
dp
q
q
p
d
q
p
q
p
d
q
q
p
Q
q
p
pq
q
p
D
q
p
Q
D
q
p
q
pq
qQ
pqQ
D
q
p
Q
D
q
p
q
pq
H
O
H'
H'
H'

20. 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. 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
2
0
0
1
0
0
2
0
2
)
1
(
)
(
,
)
1
(
)
(
)
(
)
(
0
)
(
,
]
)
(
)
(
)
(
)
(
[
t
p
q
t
q
t
p
p
t
p
t
p
t
p
t
q
dt
t
p
t
q
t
p
t
q
A T
C







 






22. 22
Toy Model - 1
qQ
Qe
e
q
Q
Qe
e
E
M
E
C
K
D
DQ
C
K
t
M
t
q
K
t
t
t
p
q
C
qp
q
p
D
DQ
q
p
t
a
a
E
t
q
t
a
a
t
p
a
qa
qp
q
p
PQP
q
p
t
p
q
t
q
t
p
p
t
p
q
dt
qp
p
q
A
D
q
i
D
q
i
iqP
iqP
C





























 















/
)
ln(
/
)
ln(
/
/
0
2
0
2
1
2
2
2
2
2
1
2
2
0
2
2
2
2
0
0
1
0
0
2
;
4
;
4
/
;
0
]
)
[(
)
(
,
)
(
)
(
)
(
:
Solution
/
,
,
:
quant.
Affine
))
(
sin(
)
/
(
)
(
,
))
(
cot(
)
(
:
Solution
2
/
;
,
,
:
quant.
Canonical
)
1
(
)
(
,
)
1
(
)
(
:
Solution
0
;
]
[
:
action.
Classical








23. 23
Toy Model - 2
)
(
)
/
(
'
'
;
|
Q
|
/
/
'
|
|
/
,
|
|
/
,
:
Model
)
|
|
,
|
|
/
(
,
)
,
(
,
)
,
(
)]
,
(
[
,
)]
,
(
/
[
,
]
)
(
;
0
[
:
action
classical
Extended
]
[
;
|)]
|
/
(
[
:
model
Toy
0
|
|
,
0
|
|
;
,
:
states
coherent
Affine
)
(
,
]
,
[
]
,
[
:
on
quantizati
Affine
2
2
2
1
2
2
1
2
2
2
2
2
1
2
2
2
1
2
2
2
1
/
|)
ln(|
/
2
1
C
s
Bohr radiu
C
me
C
P
C
e
e
C
q
C
q
C
p
q
p
Q
e
P
q
p
Q
q
p
q
P
q
p
Q
P
q
p
q
p
H
dt
q
p
H
p
q
dt
q
p
Q
P
t
i
q
p
A
a
bD
aQ
dt
q
e
p
p
q
A
q
Q
e
e
q
p
PQ
QP
D
D
Q
Q
i
Q
P
Q
m
m
m
m
R
m
C
D
q
i
ipQ













 













 




































H
H
H

24. 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. 25
TOPIC 3
• Rotationally symmetric models
• Free quantum models for
• Interacting quantum models for
• Reducible operator representation is the key
,
)
,
,
(
,
)
,
(
,
,
)
,
(
}
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,
)
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2
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q
p
Q
P
q
p
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q
p
q
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p
H
N
N
p
p
q
q
m
p
q
p
H N
n
n
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H
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
]
[ 2
p
p
p





. . .

26. 26
Rotationally Sym. Models (1)
N
that
necessary
is
it
N
When
N
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m
P
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p
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on
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coordinate
space
Phase
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H
O

27. 27
Rotationally Sym. Models (2)
;
)
(
:
Result
m)
2
/
1
(
)
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:
Uniqueness
]
1
)
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ansformati
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real
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ith
equation w
er
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m
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2
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)
cos(
2
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bp
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symmetry
rotational
full
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und state
unique gro

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
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
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








A free theory!

28. 28
…Now, Do Some Hard Work…

29. 29
Rotationally Sym. Models (3)











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N
q
q
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p
q
m
v
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p
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m
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v
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m
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;
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;
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;
,
;
,
;
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/
)
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exp[
;
,
1
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;
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;
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)
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;
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(
exp[
,
;
0
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)
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2
2
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2
4
4
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1
2
2
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2
2
2
0
2
2
1
0















































































H'
H
T
E
S
T
R
E
A
L

30. 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
q
p
q
p
q
p
H






,
,
)
,
( H

WHAT HAS BEEN ACCOMPLISHED ??

31. 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. 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)
4
n


33. 33
Main Message of Today
c
q q
c

34. 34
Thank You

35. 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