This document discusses using group theory and Lie algebras to formulate quantum mechanics from classical mechanics. It begins by reviewing classical phase space methods and their relation to Lie groups. It then develops an analogous formalism for quantum mechanics by replacing classical observables with operators satisfying the same Lie algebra. Unitary representations of this algebra define quantum states. The Heisenberg algebra is introduced for a particle, and its representation leads to a probabilistic interpretation. Dynamics are discussed using Hamiltonians of Newtonian form. As an example, the position-momentum uncertainty principle is derived from the Heisenberg commutation relation.

1. GROUP THEORY AND PARTICLE QUANTUM MECHANICS
ABSTRACT
We first review phase space methods for classical
mechanics and relate them to Lie group theory. We
proceed to develop an analogous and parallel formalism
for quantum mechanical systems by replacing classical
observables with operator-valued functions defined in a
Hilbert space. This is formally accomplished by defining
quantum observables as satisfying the same Lie algebra
as classical ones. Unitary representations of this
algebra define quantum states. Unitarity then leads the
way to a probabilistic interpretation of quantum
observables as opposed to the well defined classical
ones. We extend the discussion to Hamiltonians of
Newtonian form and single particle quantum dynamics,
providing elementary examples for concreteness.

2. Motivating Paradigm: Classical Mechanics
Classical motion of N particles determined by N 2nd
order differential equations
Trajectories fully determined by specification of
initial conditions for position and velocity
Hamilton’s Equations transform N 2nd
order
differential equations into 2N first order
differential equations in terms of phase space
variables:
( ) ( )( )
( ) ( )( )
1
1
N
N
q t q t q
p t p t p
=
=
K
K
( ),f q pdq
dt p
∂
=
∂
( ),f q pdp
dt q
∂
= −
∂

3. Motivating Paradigm: Classical Mechanics (cont.)
Solutions to Hamilton’s equations specify the orbits
of this group
( ),f q pdq
dt p
∂
=
∂
( ),f q pdp
dt q
∂
= −
∂
Such solutions leave invariant phase space volume
1 1
N
n Ndp dp dq dqω = ∧ ∧ ∧ ∧ ∧L L
under translation in time (Liouville’s theorem SM)
Π
Here is real-valued function that generates a
one-parameter group of transformations of the phase
space, denoted by . is a “classical observable”
( ),f q p
( ),f q p

4. Phase Space: Example
As a quick and simple example of phase space, we
write down the Hamiltonian for the simple harmonic
oscillator:
( )
2
21
,
2 2
p
H q p kq
m
= +
dq H p
q
dt p m
∂
= ⇒ =
∂
&
dp H
p kq
dt q
∂
= − ⇒ = −
∂
&
mq kq⇒ = −&&

5. Dynamics and Lie Algebra Structure
Canonical transformations (those which leave the
phase volume invariant) constitute a “symmetry group”
of classical mechanics
Classical and quantum mechanics can both be formulated
in terms of the process of mapping observables to one-
parameter subgroups
To construct a Lie algebra for such a system, we
first introduce a second observable and
observe its rate of change along the group generated
by
( ),g q p
( ),f q p
( ), i i
i i
i i i i
dp dqd g g
g p q
dt p dt p dt
∂ ∂
= +
∂ ∂
∑ ∑
i i i i i
g f g f
p q p p
∂ ∂ ∂ ∂
= −
∂ ∂ ∂ ∂
∑
(Inserting
Hamilton’s
equations)

6. Dynamics and Lie Algebra Structure (cont.)
We define the
Poisson bracket as:
{ },
i i i i i
g f g f
g f
p q p p
∂ ∂ ∂ ∂
≡ −
∂ ∂ ∂ ∂
∑
It can be verified that the Poisson bracket satisfies
the Jacobi identity. Accordingly, the Poisson
bracket operation converts a given set of
observables, , into a Lie algebraO
In the example above, the Hamiltonian (energy)
constituted one observable. The symmetries of the
Hamiltonian are the set of observables satisfyingf
{ }, 0f H =
Hamiltonians are usually of the form
( ) ( )21
,
2
H p q p V q= +

7. From Classical to Quantum Mechanics
One of the basic approaches to “quantizing” a
classical system is the following procedure:
•Search for unitary representations of this algebra
to define the quantum states
•Introduce quantum observables in the same form as
the classical observables, i.e., adhering to the
same Lie algebras
O
•Take the classical system with phase space
and set of real valued functions
(observables)
Π
Two objects are taken to define a quantum mechanical system:
1. A Lie algebra of observablesQ
2. A linear representation of by operators on a
Hilbert space . Each observable generates a one-
parameter group of unitary transformations on
Q
H
H

8. Definitions
1 2 2 1, ,ψ ψ ψ ψ= 1 2 1 2 1 2, , ,c c cψ ψ ψ ψ ψ ψ= =
0
k
k
ψ
∞
=
< ∞∑
, 0, 0 iff 0ψ ψ ψ≥ =
,ψ ψ ψ≡
( ) ( )1 2 1 2,
M
p p dpψ ψ ψ ψ= ∫
is also complete in the following sense:H
Example:
This sum converges in , i.e. any partial sum
converges to an element of
H
H
is a vector space over complex numbers; the
inner product between any two elements in is
Hermitian and possesses the following properties:
H
H

9. Lie Group Formalities
•The set of all unitary transformations on
constitutes a group
•This group is too large to have possess the
attractive properties of Lie groups
•We can define the Lie algebra of this group as the
set of single parameter subgroups
The Lie bracket is then defined as the group
H
With the one-parameter subgroups
we can write the “sum” of these as
( ) ,t A t→ ( ) ,t B t→
( ) ( )( )lim / /
n
n
t A t n B t n
→∞
→
( ) ( ) ( ) ( )( )
2
lim / / / /
n
n
t A t n B t n A t n B t n
→∞
→ − −
(Lie product/Trotter’s formula, corollary of BCDH)
( ) ( ) ( )
( )/ /
lim
n
A B t A t n B t n
n
e e e
+
→∞
→

10. Lie Group Formalities (cont.)
ψ(note that generator does not exist for all )
[ ] ( ),X Y XY YXψ ψ= −
( )
( )
0
lim
t
A t
X
t
ψ ψ
ψ
→
−
=
Recall that the group can also be constructed from
infinitesimal generators acting as linear operators
on . We define the generator of asH ( )t A t→
is skew-Hermitian, is HermitianX iX
These skew-Hermitian generators are operators on
corresponding to observables in . In cases of physical
interest, the “bracket” of two single parameter groups
reduces to the commutator
H
Q
We refer to each as a quantum state whose
inner product with a generator gives the expected
value of an observable for a given state
ψ

11. Particle Quantum Mechanics: Heisenberg Algebra
Let us introduce classical observables
This is the familiar nilpotent Heisenberg algebra
Since Z is the center of the algebra (commutes with
all other elements), it must be a constant multiple
of the identity. Since Z is also skew-Hermitian, it
must be purely imaginary. Accordingly,
, ,1p X q Y Z→ → →
{ } { } { }, 1, ,1 ,1 0p q q p= = =
, , 1p q
We proceed to convert these observables (real-
valued functions) to skew-Hermitian operators
{ },X Y Z ihψ ψ ψ= =
Stone-von Neumann Theorem: unitary irrep of
Heisenberg group uniquely determined by h

12. Particle Quantum Mechanics: Probability
( ) ( )q q q q dqψ ψ
∞
−∞
= ∫
( ) { }on ; on ; , on
d
q iq q p h p q ih
dq
ψ
ψ ψ ψ ψ ψ→ → − → −
With the following normalization, we can associate the
inner product with a probability and its composition
with an observable as an “expectation value”
We can represent the Heisenberg algebra as skew-
Hermitian operators on square-integrable functions
of q
( ) ( ), 1q q dqψ ψ ψ ψ
∞
−∞
= =∫
In classical mechanics, observables are real-valued
functions in phase space. In QM, the expectation value
replaces the classical observable, is a
probability measure on configuration space (observables
only have a given probability of taking on a certain
value)— hence the desire for unitary operators!
( )
2
q qψ→

13. Particle Quantum Mechanics: Dynamics
Recall: Newtonian Hamiltonians
2
2
2
d
p i
dq
→
( ) ( )
2
,
2
p
H p q V q= +
{ } { } { }2 2
, 2 ; , 2 ; ,p q p q p q pq q p= = =
( ) ( )V q iV q→
2
2
H i V
q t
ψ ψ
ψ ψ
∂ ∂
= + =
∂ ∂
As operators on square-integrable functions,
we have
Finding the one-parameter group
of unitary transformations
generated by is equivalent
to solving the Schrodinger
equation:
H
Need to extend Heisenberg algebra to functions
quadratic in (SHO: ),p q ( ) 2
V q qµ

14. Symmetries Revisited
Classical and quantum Hamiltonians possess
two types of classical symmetries:
( ) ( )
2
,
2
p
H p q V q= +
Obvious symmetries: { }, 0f V =
Extend unitary representation of the
Heisenberg algebra and to larger algebra
containing a given set of observables
H
{ }, 0f H =
{ }, 0f V ≠{ }, 0f H =Hidden symmetries:
Operators corresponding to these observables
will commute with and yield information about
energy spectrum
H

15. Computation: Obvious Symmetries
Suppose
( ) ( );
dq dp a b
a q p t
dt dt q q
∂ ∂
= = +
∂ ∂
and is linear in momentumf
Employing the “Schrodinger rules”, we posit
the following form for the quantized
where c is introduced to make f skew-Hermitian
{ }, 0f H =
( ) ( )f a q p b q= +
Hamilton’s equations become
f
d
f a ib c
dq
→ + +

16. Computation: Obvious Symmetries (cont.)
Use inner product to determine c, demanding
skew-Hermitian condition satisfied for
*
, ,a b a b
f f
f fψ ψ ψ ψ
= −
= −
1
2
da
c
dq
=
( )
( ) ( )
, a
a b a b
b
a b b a
d
f a q c dq
dt
dda
c a dq
dq dq
ψ
ψ ψ ψ
ψ
ψ ψ ψ ψ ψ
∞
−∞
∞
−∞
 
= + ÷
 
 
= − − + 
 
∫
∫
( )
1
2
d da
a q p a
dq dq
→ +

17. A Theorem
Theorem: The set of observables of the form
forms a Lie algebra under the Poisson bracket. The
representation of this algebra as a Lie algebra of
skew-Hermitian operators is defined by the assignment
Proof: Verify that this assignment is compatible
with irrep of Heisenberg algebra and Hamiltonian,
i.e. check Poisson bracket relation
( ) ( )a q p b q+
( ) ( )
1
2
d da
a q p b q a ib
dq dq
+ → + +
( ) ( )
1
2
d da
a q p b q a ib
dq dq
+ → + +

18. Proof
Classical Poisson Bracket: ( ) ( ){ } 2 1
1 2 1 2,
da da
a q p a q p a a p
dq dq
 
= − ÷
 
1 2
1 2
1 1
,
2 2
da dad d
a a
dq dq dq dq
ψ
 
+ + 
 
2 1 2
1 2 2
1 2 1
2 1 1
1 1 1
2 2 2
1 1 1
2 2 2
da da dad d d
a a a
dq dq dq dq dq dq
da da dad d d
a a a
dq dq dq dq dq dq
ψ ψ
ψ ψ
ψ ψ
ψ ψ
   
= + + + ÷  ÷
   
   
− + + + ÷  ÷
   
2 2
2 1 2 1 1 2 2 1
1 2 1 22 2
1
2
da da d a d a da da da dad
a a a a
dq dq dt dq dq dq dq dq dq
ψ
ψ
  
= − + − + − ÷ ÷
   
Observe action of commutator
on element in Hilbert space
( )
1
2
d da
a q p a
dq dq
→ +

19. Proof (cont.)
Now consider term of the form: ( ) ( ) ;b q V q V iV= →
( ) ( ){ } ( ),
V
a q p V q a q
q
∂
=
∂
( )1
,
2
d Vd da d
a iV i a aV
dq dq dq dq
dV
i a
dq
ψ ψ
ψ
ψ
  
+ = − ÷ 
   
 
=  ÷
 
Poisson bracket becomes
Observe action of commutator
on element in Hilbert space

20. Example 1: Position-Momentum Uncertainty
,q p
( ) ( )
1
2
d da
a q p b q a ib
dq dq
+ → + +
d
q iq p h
dq
→ → −
Let us take two classical observables
{ }, 1
p q p q
p q
p q q p
∂ ∂ ∂ ∂
= − =
∂ ∂ ∂ ∂
[ ] ( ) ( )
( )
( )
,p q p q q p
d d
pq qh qh
dq dq
dq
i h ih
dq
ψ ψ ψ
ψ ψ
ψ
ψ ψ
= −
= + −
= − = −

21. Example 2: Linear Oscillators
Most general system of coupled linear oscillators:
1-D Simple Harmonic Oscillator
( )2 21
2
H p q= +
,
ij i j ij i j ij i j
i j
H a q q b q p c p p= + +∑
;f p q g p q≡ + ≡ −
{ } { }
{ } { }
, ; ,
, 2; , 4
H f g H g f
f g f ig f ig i
= = −
= − + − =

22. Example 2: Linear Oscillators (cont.)
Consider unitary irrep of Lie algebra spanned by
where X,Y are operators on Hilbert space
corresponding to f,g
[ ],Y X h⇒ =
0 0 0Hu iE u=
, ,H f g
( ) ( )
1 1
; ; 1
2 2
f ig X f ig Y ih+ → − → →
Suppose H has at least one discrete eigenvalue s.t.
It follows that
( ) [ ] ( )
( )
0 0 0
0 0
,
1
H Xu H X u X Hu
i E u
= −
= +
( ) [ ] ( )
( )
0 0 0
0 0
,
1
H Yu H Y u Y Hu
i E u
= −
= −

23. Conclusion
We have reviewed how Poisson brackets determine dynamics
and symmetries of classical systems
We showed that in order to quantize a theory, we replace
the Poisson bracket with an operator valued commutator
This commutator generates the Lie algebra associated
with a given set of observables by acting on members of
a Hilbert space associated with quantum states
Finally, we uncovered the structure of a number of
operators for common Hamiltonians the most important of
which is the harmonic oscillator

24. Source
Herman, Robert; Lie Groups for Physicists, W.A.
Benjamin, INC., New York, 1966