Lie Theory From Basics to the Heisenberg Lie Group Noah Migoski IU - - PowerPoint PPT Presentation

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

Lie Theory

From Basics to the Heisenberg Lie Group Noah Migoski

IU Math DRP

April, 2020

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

What is a Group?

From Artin [1]: ”A group is a set G together with a law of composition (a map from two elements to another element in the set) that has the following properties”: The law of composition is associative: (ab)c = a(bc) for all a, b, c in G.

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

What is a Group?

From Artin [1]: ”A group is a set G together with a law of composition (a map from two elements to another element in the set) that has the following properties”: The law of composition is associative: (ab)c = a(bc) for all a, b, c in G. G contains an identity element 1, such that 1a = a and a1 = a for all a in G.

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

What is a Group?

From Artin [1]: ”A group is a set G together with a law of composition (a map from two elements to another element in the set) that has the following properties”: The law of composition is associative: (ab)c = a(bc) for all a, b, c in G. G contains an identity element 1, such that 1a = a and a1 = a for all a in G. Every element a of G has an inverse, an element b such that ab = 1 and ba = 1.

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

Examples of Groups

Examples Z+: The set of integers with additon as its law of composition.

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

Examples of Groups

Examples Z+: The set of integers with additon as its law of composition. R+, R×, C+, C×: The real or complex numbers under addition or multiplication (with zero removed).

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

Examples of Groups

Examples Z+: The set of integers with additon as its law of composition. R+, R×, C+, C×: The real or complex numbers under addition or multiplication (with zero removed). Cyclic Groups: The rotational symmetry of a regular polygon

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

Examples of Groups

Examples Z+: The set of integers with additon as its law of composition. R+, R×, C+, C×: The real or complex numbers under addition or multiplication (with zero removed). Cyclic Groups: The rotational symmetry of a regular polygon The Dihedral Groups: The symmetries of a regular polygon including reflections.

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

Examples of Groups

Examples Z+: The set of integers with additon as its law of composition. R+, R×, C+, C×: The real or complex numbers under addition or multiplication (with zero removed). Cyclic Groups: The rotational symmetry of a regular polygon The Dihedral Groups: The symmetries of a regular polygon including reflections. The symmetric groups: The permutations of a set of n elements

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

What is a Lie Group?

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

What is a Lie Group?

“A Lie group is, roughly speaking, a continuous group” [1]. A more concrete definition is the following:

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

What is a Lie Group?

“A Lie group is, roughly speaking, a continuous group” [1]. A more concrete definition is the following: Lie Group Definition A group (which is also a manifold) G such that the group product G × G → G and the inverse map G → G are smooth. Meaning they are infinitely differentiable.

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

Examples

U(n): The unitary groups. (n by n complex matricies satisfying: X ∗X = XX ∗ = I)

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

Examples

U(n): The unitary groups. (n by n complex matricies satisfying: X ∗X = XX ∗ = I) O(n): The orthogonal groups. (n by n real matricies satisfying: X TX = XX T = I)

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

Examples

U(n): The unitary groups. (n by n complex matricies satisfying: X ∗X = XX ∗ = I) O(n): The orthogonal groups. (n by n real matricies satisfying: X TX = XX T = I) SU(n) and SO(n): The untitary or orthogonal groups with the added condition that det(X) = 1.

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

What is a Lie Algebra?

Lie Algebra Definition A Lie algebra is a finite-dimensional real or complex vector space g, together with a map[·, ·] from g × g → g, satisfying: [·, ·] is bilinear. A Lie algebra can be thought of as the tangent space at the identity of a Lie group. They are connected by the exponential map.

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

What is a Lie Algebra?

Lie Algebra Definition A Lie algebra is a finite-dimensional real or complex vector space g, together with a map[·, ·] from g × g → g, satisfying: [·, ·] is bilinear. [·, ·] is skew symmetric: [X, Y ] = −[Y , X]∀X, Y ∈ g A Lie algebra can be thought of as the tangent space at the identity of a Lie group. They are connected by the exponential map.

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

What is a Lie Algebra?

Lie Algebra Definition A Lie algebra is a finite-dimensional real or complex vector space g, together with a map[·, ·] from g × g → g, satisfying: [·, ·] is bilinear. [·, ·] is skew symmetric: [X, Y ] = −[Y , X]∀X, Y ∈ g The Jacobi identity holds: [X, [Y , Z]] + [Y , [Z, X]] + [Z, [X, Y ]] = 0 A Lie algebra can be thought of as the tangent space at the identity of a Lie group. They are connected by the exponential map.

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

The Exponential Map

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

The Exponential Map

Recall the Taylor series for ex: ex =

∞

• m=0

xm m!

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

The Exponential Map

Recall the Taylor series for ex: ex =

∞

• m=0

xm m! It can be shown that this sum also converges when x is an n × n matrix sufficiently close to the identity.

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

The Exponential Map

Recall the Taylor series for ex: ex =

∞

• m=0

xm m! It can be shown that this sum also converges when x is an n × n matrix sufficiently close to the identity. If G is a matrix Lie group with Lie algebra g, then the exponential map for G is the map exp : g → G : X → etX for t ∈ R

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

Representations

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

Representations

A representation can be thought of as a linear action of a group or Lie algebra on a vector space. Representations as matricies allow us to more easily do computations in order to observe the properties of more abstract algebraic structures.

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

Representations

A representation can be thought of as a linear action of a group or Lie algebra on a vector space. Representations as matricies allow us to more easily do computations in order to observe the properties of more abstract algebraic structures. Representation Definition Let G be a group. A representation of G is a homomorphism Π : G → GL(V ) Where V is a vector space.

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

Representations

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

Representations

There is a natural connection between homomorphisms in Lie groups and homomorphisms in their corresponding Lie algebras. Because of this the representations of a Lie group tell us about the representations of its Lie algebra.

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

Representations

There is a natural connection between homomorphisms in Lie groups and homomorphisms in their corresponding Lie algebras. Because of this the representations of a Lie group tell us about the representations of its Lie algebra. In physics we are concerned with describing physical laws. Representation theory allows us to derive differential equations for these laws based on the symmetries of the system we are describing.

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

R and Translations in Time

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

R and Translations in Time

In physics we want to find how states ϕ change with time. That is, finding an ”equation of motion” that allows us to calculate future states ϕ(t) from an initial state ϕ(0). To find such an equation we will consider an operator U(t) that moves our state forward in time.

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

R and Translations in Time

In physics we want to find how states ϕ change with time. That is, finding an ”equation of motion” that allows us to calculate future states ϕ(t) from an initial state ϕ(0). To find such an equation we will consider an operator U(t) that moves our state forward in time. ϕ(t) = U(t)ϕ(0)

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

R and Translations in Time

In physics we want to find how states ϕ change with time. That is, finding an ”equation of motion” that allows us to calculate future states ϕ(t) from an initial state ϕ(0). To find such an equation we will consider an operator U(t) that moves our state forward in time. ϕ(t) = U(t)ϕ(0) When our state space is finite dimensional, U(t) is a differentiable unitary representation of R on the state space.

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

R and Translations in Time

In physics we want to find how states ϕ change with time. That is, finding an ”equation of motion” that allows us to calculate future states ϕ(t) from an initial state ϕ(0). To find such an equation we will consider an operator U(t) that moves our state forward in time. ϕ(t) = U(t)ϕ(0) When our state space is finite dimensional, U(t) is a differentiable unitary representation of R on the state space. U(t) = et −i

H Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

R and Translations in Time

In physics we want to find how states ϕ change with time. That is, finding an ”equation of motion” that allows us to calculate future states ϕ(t) from an initial state ϕ(0). To find such an equation we will consider an operator U(t) that moves our state forward in time. ϕ(t) = U(t)ϕ(0) When our state space is finite dimensional, U(t) is a differentiable unitary representation of R on the state space. U(t) = et −i

H

With a corresponding Lie Algebra representation with elements

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

R and Translations in Time

In physics we want to find how states ϕ change with time. That is, finding an ”equation of motion” that allows us to calculate future states ϕ(t) from an initial state ϕ(0). To find such an equation we will consider an operator U(t) that moves our state forward in time. ϕ(t) = U(t)ϕ(0) When our state space is finite dimensional, U(t) is a differentiable unitary representation of R on the state space. U(t) = et −i

H

With a corresponding Lie Algebra representation with elements −i H = d dt = ⇒ H = i ∂ ∂t

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

R3 and Translations in Space

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

R3 and Translations in Space

Most states depend not only on time but also on their location in space − → q = (q1, q2, q3).To describe translations in space, we will define the corresponding Lie algebra representations using self-adjoint operators P1, P2, P3 that play the same role for spatial translations that H plays for time translations.

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

R3 and Translations in Space

Most states depend not only on time but also on their location in space − → q = (q1, q2, q3).To describe translations in space, we will define the corresponding Lie algebra representations using self-adjoint operators P1, P2, P3 that play the same role for spatial translations that H plays for time translations. Momentum operators [2] For a system with state space given by complex-valued functions of position variables q1, q2, q3 the momentum operators P1, P2, P3 are defined by P1 = −i ∂ ∂q1 , P2 = −i ∂ ∂q2 , P3 = −i ∂ ∂q3 It can be convinient to write these as a single operator − → P = −i− → ∇

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

The Free Particle 1

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

The Free Particle 1

You may remember from introductory physics that the (non-relativistic) equation for kinetic energy is:

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

The Free Particle 1

You may remember from introductory physics that the (non-relativistic) equation for kinetic energy is: E = 1

• 2mv2. And for momentum is p = mv. Combining these gives:

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

The Free Particle 1

You may remember from introductory physics that the (non-relativistic) equation for kinetic energy is: E = 1

• 2mv2. And for momentum is p = mv. Combining these gives:

E = p2 2m

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

The Free Particle 1

You may remember from introductory physics that the (non-relativistic) equation for kinetic energy is: E = 1

• 2mv2. And for momentum is p = mv. Combining these gives:

E = p2 2m Replacing E and p with their corresponding operators gives:

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

The Free Particle 1

You may remember from introductory physics that the (non-relativistic) equation for kinetic energy is: E = 1

• 2mv2. And for momentum is p = mv. Combining these gives:

E = p2 2m Replacing E and p with their corresponding operators gives: H = |− → P |2 2m

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

The Free Particle 2

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

The Free Particle 2

This tells us that states ϕ(− → q , t) in our system obey:

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

The Free Particle 2

This tells us that states ϕ(− → q , t) in our system obey: Hϕ(− → q , t) = |− → P |2 2m ϕ(− → q , t)

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

The Free Particle 2

This tells us that states ϕ(− → q , t) in our system obey: Hϕ(− → q , t) = |− → P |2 2m ϕ(− → q , t) From our representations of H and − → P we have: i ∂ ∂t ϕ(− → q , t) = −2 2m ∇2ϕ(− → q , t) Which is the Schr¨

• dinger equation for a free particle.

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

The Free Particle 3

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

The Free Particle 3

The general solution to this equation is: |ϕ(− → q , t) =

• −

→ k

c−

→ k ϕ(−

→ q )ϕ(t) =

• −

→ k

c−

→ k ei− → k ·− → q e−it |−

→ k |2 2m

Where − → p = − → k .

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

The Free Particle 3

The general solution to this equation is: |ϕ(− → q , t) =

• −

→ k

c−

→ k ϕ(−

→ q )ϕ(t) =

• −

→ k

c−

→ k ei− → k ·− → q e−it |−

→ k |2 2m

Where − → p = − → k . Notice that we are expressing a general state in terms of a sum (or superposition) of states defined by their value of − → k . These states are linearly independent, so we can think of the genearal state as a vector in a state space. Which we denote with the |ϕ or ”ket” symbol.

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

Eigenvalues

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

Eigenvalues

Notice that each of the terms in this sum are eigenfunctions of − → P and H with eigenvalues − → p and E respectively.

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

Eigenvalues

Notice that each of the terms in this sum are eigenfunctions of − → P and H with eigenvalues − → p and E respectively. − → P |ϕ(− → q , t)−

→ k = −

→ p |ϕ(− → q , t)−

→ k , and H|ϕ(−

→ q , t)−

→ k = E|ϕ(−

→ q , t)−

→ k

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

Eigenvalues

Notice that each of the terms in this sum are eigenfunctions of − → P and H with eigenvalues − → p and E respectively. − → P |ϕ(− → q , t)−

→ k = −

→ p |ϕ(− → q , t)−

→ k , and H|ϕ(−

→ q , t)−

→ k = E|ϕ(−

→ q , t)−

→ k

Similarly, we can define an operator Q who’s eigenvalues are the position of the state.

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

Eigenvalues

Notice that each of the terms in this sum are eigenfunctions of − → P and H with eigenvalues − → p and E respectively. − → P |ϕ(− → q , t)−

→ k = −

→ p |ϕ(− → q , t)−

→ k , and H|ϕ(−

→ q , t)−

→ k = E|ϕ(−

→ q , t)−

→ k

Similarly, we can define an operator Q who’s eigenvalues are the position of the state. Q|ϕ(− → q , t)−

→ k = −

→ q |ϕ(− → q , t)−

→ k

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

The Heisenberg Lie Algebra

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

The Heisenberg Lie Algebra

Heisenberg Lie Algebra Definition (d = 1) [2] The Heisenberg Lie algebra h3 is the vector space R3 with the Lie bracket defined by its values on a basis (X,Y,Z) by [X, Y ] = Z, [X, Z] = [Y , Z] = 0 This is isomorphic to the Lie algebra of 3 by 3 strictly upper triangular real matricies, with Lie bracket the matrix commutator: X →   1   , Y →   1   , Z →   1  

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

The Heisenberg Lie Algebra in higher dimensions

Heisenberg Lie Algebra Definition (higher dimensions) [2] The Heisenberg Lie algebra h2d+1 is the vector space R2d+1 = R2d ⊕ R with the Lie bracket defined by its values on a basis (Xj, Yk, Z) for j, k < d by [Xj, Yk] = δjkZ, [Xj, Z] = [Yk, Z] = 0 For physics we are primarily concerned with the case of d = 3, for which a general element can be written as a matrix:       x1 x2 x3 z y1 y2 y3      

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

The Heisenberg Lie Algebra and Quantum Mechanics

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

The Heisenberg Lie Algebra and Quantum Mechanics

Consider the following calculation of the Lie bracket of the P and Q operators we’ve constructed.

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

The Heisenberg Lie Algebra and Quantum Mechanics

Consider the following calculation of the Lie bracket of the P and Q operators we’ve constructed. [P, Q]f = (QP − PQ)f = QPf − PQf = Q(−i d dq )f + (−i d dq )Qf = −iq df dq − i d dq (qf ) = −iq df dq − i(f + q df dq ) = −i(q df dq − f − q df dq ) = if (1)

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

The Heisenberg Lie Algebra and Quantum Mechanics

Consider the following calculation of the Lie bracket of the P and Q operators we’ve constructed. [P, Q]f = (QP − PQ)f = QPf − PQf = Q(−i d dq )f + (−i d dq )Qf = −iq df dq − i d dq (qf ) = −iq df dq − i(f + q df dq ) = −i(q df dq − f − q df dq ) = if (1) = ⇒ [P, Q] = i We can use these to construct a representation of the Heisenberg Lie algebra.

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

The Schr¨

• dinger Representation

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

The Schr¨

• dinger Representation

Schr¨

• dinger representation, Lie algebra version [2]

The Schr¨

• dinger representation of the Heisenberg Lie algebra h is

the representation (Γ′

S, L2(R)) satisfying

Γ′

S(X)ϕ(q) = Qϕ(q) = qϕ(q)

Γ′

S(Y )ϕ(q) = Pϕ(q) = −i d

dq ϕ(q) Γ′

S(Z)ϕ(q) = iϕ(q)

(2)

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

Commutation Relations

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

Commutation Relations

Consider [H,P] for a free particle:

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

Commutation Relations

Consider [H,P] for a free particle: [H, P]f = [P2/2m, P]f = P2/2mPf − PP2/2mf = (−i d

dq)2

2m ( −i d

dq

2m )f − ( −i d

dq

2m ) (−i d

dq)2

2m f = −2 4m2 ((d3f dq3 − d3f dq3 ) = 0

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

Commutation Relations

Consider [H,P] for a free particle: [H, P]f = [P2/2m, P]f = P2/2mPf − PP2/2mf = (−i d

dq)2

2m ( −i d

dq

2m )f − ( −i d

dq

2m ) (−i d

dq)2

2m f = −2 4m2 ((d3f dq3 − d3f dq3 ) = 0 = ⇒ [H, P] = 0

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

Importance

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

Importance

[H, P] = 0 tells us that P does not change with time, so momentum is a conserved quantity.

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

Importance

[H, P] = 0 tells us that P does not change with time, so momentum is a conserved quantity. It can also be shown that [H, Q] = 0 implying position is not a conserved quantity, which is what we would expect for a free particle.

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

Importance

[H, P] = 0 tells us that P does not change with time, so momentum is a conserved quantity. It can also be shown that [H, Q] = 0 implying position is not a conserved quantity, which is what we would expect for a free particle. Since [P, Q] = 0. we can use this to show that this implies that states with both well-defined position and well-defined momenum do not exist.

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

The state space of a quantum particle, either free or moving in a potential, will be a unitary representation of the Heisenberg Lie group, with the group of spatial translations as a subgroup. This group is fundemental to the structure of quantum mechanics as the noncommutativity of P and Q accounts for the innability to measure both the exact momentum and position of a quantum particle.

Noah Migoski Lie Theory

Introduction Representation Theory Applications to Physics The Heisenberg Lie Algebra

References

1 Algebra 2nd edition, Michael Artin 2 Lie Groups, Lie Algebras, and Representations 2nd edition, Brian C. Hall 3 Quantum Theory, Groups and Representations, Peter Woit

Noah Migoski Lie Theory