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."
8. 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
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
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
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i
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A
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t
p
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dt
t
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))]
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)
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(
:
action
New
)
(
),
(
)
(
:
variation
Restricted
)
(
]
/
[
)
(
:
action
Quantum
H
H
CLASSICAL MECHANICS IS QUANTUM
MECHANICS RESTRICTED TO A CERTAIN TWO
DIMENSIONAL SURFACE IN HILBERT SPACE
subset
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
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 !
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
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1
(
)
(
,
)
1
(
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0
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(
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[
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
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K
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t
t
p
q
C
qp
q
p
D
DQ
q
p
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a
a
E
t
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t
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a
qa
qp
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p
PQP
q
p
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p
q
t
q
t
p
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t
p
q
dt
qp
p
q
A
D
q
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D
q
i
iqP
iqP
C
/
)
ln(
/
)
ln(
/
/
0
2
0
2
1
2
2
2
2
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1
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0
2
2
2
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0
1
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0
2
;
4
;
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)
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Solution
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quant.
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sin(
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cot(
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quant.
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1
(
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(
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Solution
0
;
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action.
Classical
23. 23
Toy Model - 2
)
(
)
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(
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;
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Model
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model
Toy
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states
coherent
Affine
)
(
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on
quantizati
Affine
2
2
2
1
2
2
1
2
2
2
2
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2
2
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1
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|)
ln(|
/
2
1
C
s
Bohr radiu
C
me
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P
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q
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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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)
,
( 1
2
2
0
2
2
0
2
2
1
q
p
Q
P
q
p
q
p
H
q
p
q
p
q
p
H
N
N
p
p
q
q
m
p
q
p
H N
n
n
H
H
]
[ 2
p
p
p
. . .
26. 26
Rotationally Sym. Models (1)
N
that
necessary
is
it
N
When
N
Q
Q
m
P
i
P
Q
Q
q
P
p
Y
XZ
q
p
L
E
q
Z
q
p
Y
p
X
R
N
O
q
O
q
p
O
p
N
q
q
m
p
q
p
H
q
q
q
p
p
p
jk
k
j
N
N
/
,
,
:
)
(
:
:
:
:
n
Hamiltonia
]
,
[
;
,
:
on
Quantizati
)
(
,
:
motion
of
Constants
,
,
:
invariants
Basic
)
,
(
;
,
under
Invariant
;
)
(
]
[
)
,
(
)
,...,
(
,
)
,...,
(
:
s
coordinate
space
Phase
0
2
2
0
2
2
0
2
2
1
2
2
2
2
2
2
2
0
2
2
0
2
2
1
1
1
H
O
27. 27
Rotationally Sym. Models (2)
;
)
(
:
Result
m)
2
/
1
(
)
(
:
Uniqueness
]
1
)
(
[
;
)
(
)
(
)
sin(
)
(
)
(
)
(
on
distributi
state
ground
the
of
on
ansformati
Fourier tr
).
(
)
(
:
with
)
(
real
a
ith
equation w
er
Schroeding
m
4
/
0
0
2
/
2
1
2
)
2
(
2
/
2
2
1
2
/
)
cos(
2
/
2
2
2
2
2
p
bp
N
N
N
N
r
p
N
N
N
N
N
ipr
N
x
p
i
N
N
N
N
e
p
C
b
b
f
db
b
f
db
b
f
e
d
dr
r
r
e
K
d
d
dr
r
r
e
x
d
x
e
p
C
x
r
symmetry
rotational
full
x
und state
unique gro
A free theory!
29. 29
Rotationally Sym. Models (3)
N
q
q
m
p
q
m
v
q
m
q
m
p
q
p
Q
S
m
R
v
Q
S
m
R
S
Q
m
P
q
p
q
p
q
p
P
q
Q
p
i
q
p
R
i
Q
S
m
P
i
S
Q
m
N
q
m
p
w
q
m
p
q
p
Q
m
P
w
Q
m
P
q
p
q
p
q
p
P
q
Q
p
i
q
p
P
i
Q
m
;
)
(
)
(
)
(
)
(
;
,
:}
]
)
(
[
:
:
)
(
:
:
)
(
:
{
;
,
;
,
;
,
;
0
]
/
)
(
exp[
;
,
1
0
;
0
;
0
]
)
(
[
;
0
]
)
(
[
;
)
(
)
(
,
:}
)
(
:
:
:
{
,
,
~
,
0
]
/
)
(
exp[
,
;
0
0
)
(
2
2
0
2
2
0
2
2
1
2
2
4
4
2
2
2
2
1
2
2
2
2
1
2
2
2
2
2
2
2
2
1
2
2
2
2
1
2
2
2
0
2
2
2
0
2
2
1
2
2
2
0
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