Symmetry and Action Principles in Physics

Tom Charnock

Contents

1 Classical Mechanics 2

1.1 Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Canonical Transformations 4

2.1 Phase Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1.1 Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Canonical Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2.1 Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3 Continuous Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3.1 Coordinate Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3.2 Translational Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.4 Rotational Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Poissons Brackets 9

3.1 Time Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.2 Spacial Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4 Noethers Theorem 10

4.1 Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4.1.1 Time Translation Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4.1.2 Spacial Translation Invariance . . . . . . . . . . . . . . . . . . . . . . . . . 11

4.2 Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4.2.1 Scalar Phase-Space Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4.2.2 Hamiltonian Phase-Space Function . . . . . . . . . . . . . . . . . . . . . . . 11

5 Finite Transformations 12

5.1 Infinitesimal Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

5.2 Finite Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

5.3 Group Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

5.4 Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5.5 Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5.5.1 Translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5.5.2 Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5.6 Combining Finite Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5.7 Newtonian Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

5.7.1 Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

5.7.2 Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

5.7.3 Galilean Boost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

5.7.4 Space-time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2

6 Statistical Mechanics 156.1 Canonical Ensemble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156.2 Statistical Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

6.2.1 Helmholtz Free Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156.2.2 Partition Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156.2.3 Gibbs Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

6.3 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156.4 Phase Transistions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166.5 Magnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

6.5.1 Low Temperature Probability . . . . . . . . . . . . . . . . . . . . . . . . . . 176.5.2 High Temperature Probability . . . . . . . . . . . . . . . . . . . . . . . . . 176.5.3 Other Temperature Probability . . . . . . . . . . . . . . . . . . . . . . . . . 17

6.6 Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186.6.1 1D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186.6.2 2D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186.6.3 3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

7 Relativity and Space-Time 207.0.4 Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

7.1 The Invariant Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207.2 Lorentz Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

8 Electromagnetism 228.1 Electromagentic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228.2 Sourced Maxwell Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

8.2.1 Gauss’ Law for Electric Fields . . . . . . . . . . . . . . . . . . . . . . . . . . 228.2.2 Ampre’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

8.3 Maxwell Equations for Electromagnetism . . . . . . . . . . . . . . . . . . . . . . . 238.3.1 Field Strength Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238.3.2 Gauss’ Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238.3.3 Ampre’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

8.4 Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

9 Relativistic Actions 249.1 Relativistic Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1

Classical Mechanics

A system with two identical masses attached by a spring has a potential V = 12k|r˜A − r˜B |2 and

kinetic energy T = 12m(r˜2 + r˜2).

1.1 Lagrangian

The Lagrangian is therefore:

L =1

2m(r˜2 + r˜2)− 1

2k|r˜A − r˜B |2

This can be replaced by the centre of mass coordinates R˜ =r˜A+r˜B2 and % = r˜A − r˜B so that the

Lagrangian becomes:

L =1

2mR˜2 +

1

2m%˜2 − 1

2%˜2

R is a cyclic coordinate, which indicates the conservation of the total momentum. If % = %(cos θ, sin θ)then the angular kinetic energy is T = 1

2m%˜2 + 12m%˜2θ˜2 and so θ can be seen to be a cyclic co-

ordinate, which indicates there are no angular forces. There are three degrees of freedom in thissystem, R, θ and %. The Euler-Lagrange equations can be used to show that momenta is conserved.

δ

δt

∂L∂q− ∂L∂q

= 0

The conservation of linear momentum can be seen by MR˜ = p˜R = 0 and the conservation of

angular momentum can be seen by δδt

(m%2θ

)= p˜θ = 0. The equation of motion of the vibration

of the spring is:

% = %θ2 − k

m%

1.2 Hamiltonian

The Hamiltonian is given by:

H =∑β

pβ qβ − L

The generalised momenta are:

pR = mR

pθ = m%2θ

p% = m%

2

The Hamiltonian of this system is:

H =p2r2

+p2θm%2

+p2%m− L

Hamilton’s equations can then be calculated to find the equation of motion. Hamilton’s equationsare ∂H

∂pα= qα and ∂H

∂qα= −pα.

3

2

Canonical Transformations

2.1 Phase Space

If there are N degrees of freedom then there are N generalised coordinates q1, q2, ..., qN . The spacespanned by

{q}

is called configuration space, where the dimension is N . For one degree of freedomthe configuration space is:

0 q

Figure 2.1: Configuration Space

The space spanned by{q.p}

is called phase space, where the dimension is 2N . q and p can beindependent in some cases. A point in phase space (q, p) is a state of the system. When theHamiltonian H(q, p) is known, the whole evolution from that point can be deduced.

2.1.1 Harmonic Oscillator

The Hamiltonian for a harmonic oscillator is:

H =p2

2m+kq2

2

The energy is conserved as H = 0 and so 2mE = p2 +mkq2 which is the equation for an ellipse inphase space.

q

p

(q, p)

Figure 2.2: Phase-Space

Phase-space orbits cannot cross. If this were to happen then there would be a choice in the system,but phase-space must be totally deterministic.

4

2.2 Canonical Transformations

A transformation of the Hamiltonian creates a new function, which might not be a Hamiltonian.If the transformations create a new Hamiltonian then the transformation is canonical.

H(q, p)→ K(q′, p′)

The Hamiltonian of a harmonic oscillator is H = p2

2m + kq2

2 then transforming the coordinates

from q′ = ±p and p′ = q gives Hamilton’s equations as, q′ = ∂K∂p′ = kp′ and p′ = −∂K∂q′ = − q′

m .

This means that for q′ = −p the correct equations of motion are recovered. This is a canonicaltransformation where K is a Hamiltonian. For the harmonic oscillator the variables q,p,q′ and p′

are cyclic variables.

2.2.1 Generating Functions

Canonical transformations can be obtained from a single function called the generating function.The generating function which transforms (q, p) → (q′p′) must have half of the information fromthe original function and half from the new function. If F1(q, q′) = −qq′ then p = ∂F1

∂q = −q′ and

p′ = −∂F1

∂q′ = q. There are four possible generating functions:

F1(q, q′)→ p =∂F1

∂q, p′ = −∂F1

∂q′

F2(q, p′)→ p =∂F2

∂q, q′ =

∂F2

∂p′

F3(p, q′)→ q = −∂F3

∂q, p′ = −∂F3

∂q′

F4(p, p′)→ q =∂F4

∂p, q′ = −∂F4

∂p′

2.3 Continuous Symmetries

2.3.1 Coordinate Transformations

A Lagrangian L(q, q) is transformed to a position of q = q(q′) and a velocity of q = q(q′, q′) tobecome L(q′, q′) which is a new function of the new variables. When L(q, q) = L(q′, q′) then thereis a symmetry.

m

k

Figure 2.3: Mass Moving in a Potential

L =1

2m(x2 + y2)− 1

2(x2 + y2)

When this system is transformed into polar coordinates such that x = r cos θ and y = r sin θ thenthe Lagrangian becomes:

L′ =1

2m(r2 + r2θ2)− 1

2r2

As the Lagrangians are different there is no symmetry.

5

2.3.2 Translational Symmetry

The translation of a point is given by r˜ = r˜′ − a˜ and can be interpreted in two forms, Passive andActive.

x′

xa˜

y′

y

r˜′ r˜

Figure 2.4: Passive Translation

Passive translation occurs when the object remains stationary, but the frame of reference changes.The origin moves by a˜. For a harmonic oscillator the phase-space orbit is circular when the energyis conserved, and for a passive change in phase-space, q′ = q − α and p′ = p − β. A point inphase-space A becomes A′ under this passive transformation, which remains on the same energycontour.

q′

A′

q′β

A q

α

p′

p p′

Figure 2.5: Passive Translation in Phase-Space

The point A′ is given by q′ = qmax−α and p′ = −β. Active translation occurs where the frame ofreference remains constant and the object translates.

x

y

a˜r˜′

r˜Figure 2.6: Active Translation

6

For an active change of a harmonic oscillator in phase-space, q′ = q−α and p′ = p− β. The pointA translates to B, which has a different energy and so denotes a physical change to the system.

q

p

A

Bαβ

Figure 2.7: Active Translation in Phase Space

δU = U(B)− U(A′)

If a system has translational symmetry then the Lagrangian must remain unchanged under atranslation.

Composition

More than one translation can be made such that translating r + Tα = r′ then r′ + Tβ = r′′ is acomposition r + Tα+β = r′′.

Inverse

For every operation there is an inverse operation such that the translation r + Tα = r′ has theinverse r′ + T−α = r.

Identity

The composition of a translation and its inverse is the identity so r + Tα−α = r.

2.4 Rotational Symmetry

A rotation can be described by a rotation matrix:

r˜ =

(cos θ − sin θsin θ cos θ

)· r˜′ = R(θ) · r˜′ (2.1)

r˜r˜′

θ

x

y

Figure 2.8: Rotation

7

The rotation matrix is antisymmetric, where the eigenvalues are eiθ and e−iθ and |(θ)| = 1.

RT (θ) ·R(θ) = 1

If a Lagrangian is unchanged under rotation then there can be two kinds of symmetry. Continuoussymmetry can exist if θ is unrestricted such that θ[0; 2π], or discrete symmetry if θ is restricted.

Composition

Several rotations can be made to make a composition in the same way as for translations. r˜·R(α) =r˜′ then r˜′ ·R(β) = r˜′′, can be written as r˜ ·R(α+ β) = r˜′′.Inverse

The inverse of a rotation can be seen from r˜ ·R(α) = r˜′ has an inverse r˜′−R = r˜Identity

r˜ · R−R = r˜ is the identity. This is the same as taking the product of the transform and therotation matrix.

8

3

Poissons Brackets

Poisson Brackets are algebraic properties of classical mechanics in the Hamiltonian formalism.They are direct connections between classical mechanics and quantum mechanics. Poisson Bracketsfor position and momentum are called Canonical (Fundamental) Brackets.{

f(qα, pα), g(qα, pα)

}=∑α

[∂f

∂qα

∂g

∂pα− ∂g

∂qα

∂f

∂pα

]{qi, pj

}= δij and

{qi, qj

}={pi, pj

}= 0. The Poisson bracket is closely related to the commutator

in quantum mechanics because: {qα, pα

}=⇒

[qα, pα

]= ih

The Poisson bracket is antisymmetric{f, g}

= −{g, f}

, which also shows that{f, f}

= 0. It is

also linear such that{af + bg, h

}= a

{f, h}

+ b{g, h}

. The Leibneitz rule is a derivative product

rule which shows that{fg, h

}= f

{g, h}

+{f, h}g. The antisymmetric property also defines

the Jacobi identity where{f,{g, h}}

+{h{f, g}}

+{g,{h, f

}}= 0. The quantum mechanical

commutator has all the same properties.

3.1 Time Evolution

The Hamiltonian can be said to generate time evolution as δfδt = ∂f

∂t + q ∂f∂q + p∂f∂p which, when fis independent of time then:

δf

δt={f,H

}When the Hamiltonian is independent of time then energy is conserved.{

H,H}

=δHδt

= 0

If the components of the momentum are constant in time then{p,H

}= 0.

3.2 Spacial Evolution

The momentum is the generator of spacial transformations such that{f, p}

= ∂f∂q . The angular

momentum is the generator of spacial rotations.

9

4

Noethers Theorem

4.1 Lagrangian

If coordinates are set up such that t = t + εξ(t) and q(t) = q(t) + εη(q, t) then for an invariantsystem: ∫ t2

t1

δtL(q(t),

δq(t)

δt, t

)=

∫ t2

t1

δt

[L(q, q, t) + ε

δGδt

]Substituting the coordinates in and expanding to O(ε) gives:

L(q(t),

δq(t)

δt, t

)= L

(q + εη, q + ε(η − ξq), t+ εξ

)When this is Taylor expanded and placed back in the invariance equation then:

ε

∫ t2

t1

δt

[Lξ + ξ

∂L∂t

+ η∂L∂q

+ (η + ηξq)∂L∂q− δGδt

]= 0

As this holds for all time then the integral can be cancelled and using the total time derivative ofL this expands to:

δ

δt

[ξL+ (η − ξq)∂L

∂q− G

]− (η − ξq)

[δ

δt

(∂L∂q

)− ∂L∂q

]= 0

As the last term is the Euler-Lagrange equations, which are equal to zero, then this gives:

ξL+ (η − ξq)∂L∂q− G = constant

This is Noether’s Theorem.

4.1.1 Time Translation Invariance

If the time and space coordinates are given by t = t + to and q(t) = q(t) then this means ξ isconstant and η = 0. For a Lagrangian L(x, x) = 1

2mx2 − V (x) then:

∫ t2

t1

δt

[1

2m ˙x2 − V (x)

]=

∫ t2

t1

δt

[1

2mx2 − V (x)

]This shows that G = 0 and so Nther’s Theorem gives L − q ∂L∂q is a constant. This is the negativeof the Hamiltonian, and shows that a translation in time has a constant Hamiltonian.

10

4.1.2 Spacial Translation Invariance

When the time coordinates are given by t = t and the space coordinates are given by q(t) = q(t)+Qthen ξ = 0 and η is constant. For a system with no potential the Lagrangian is L = 1

2mx2 and so

for invariance. ∫ t2

t1

δtL(x) =

∫ t2

t1

δtL(x)

So G = 0 and therefore Nether’s Theorem shows η ∂L∂q is constant, which is the conjugate momentum.

4.2 Hamiltonian

For the generating function F2 the transformations are p′ = ∂F2

∂p′ , p = ∂F2

∂q and H ′ = H + ∂F2

∂t . Togenerate small changes then the generating function must be small. The identity transformation is

F(id)2 = qp′ where the momentum, position and Hamiltonian are unchanged under transformation.

Close to the identity is a generating function for small changes.

F2 = qp′ + εG(q, p′, t)

This gives the transformation p = p′+ ε∂G∂q , q′ = q+ ε ∂G∂p′ and H′ = H+ ε∂G∂t . To the leading order

of O(ε), G can be considered to be G(q, p, t) rather than G(q, p′, t). This gives:

δq = q′ − q = ε∂G∂p

= ε{q,G}

δp = p′ − p = ε∂G∂q

= ε{p,G

}4.2.1 Scalar Phase-Space Functions

A scalar function is f ′(q′) = f(q) where q and q′ are the coordinates of the observers positions Oand O′. f and f ′ are the values of the functions measured at O and O′. Scalar functions can bedefined U ′(A′) = U(A) so δU = U(B)− U(A′) = U(B)− U(A). Using Taylor Expansion:

δU = U(q, p) + δq∂U

∂q+ δp

∂U

∂p− U(q, p)

Substituting in the values of δq and δp gives:

δU = ε{U,G

}4.2.2 Hamiltonian Phase-Space Function

As the Hamiltonian is not a scalar function:

H′(A′) = H(A) + ε∂G∂t

δH = H(B)−H(A)− ε∂G∂t and so, using Taylor expansion δH = ε{H,G

}− ε∂G∂t . This is the total

derivative of the generating function with respect to time.

δH = −εδGδt

If the Hamiltonian is unchanged then G is conserved. When G = p the generator is that of spacetransformations δq = ε

{q, p}

= ε.

11

5

Finite Transformations

5.1 Infinitesimal Transformations

For infinitesimal time transformation t → t + δt =⇒ f(t) → f(t + δt) which, due to Taylorexpansion, is:

f(t+ δt) ≈ f(t) + δtδf(t)

δt+ ... = f(t) + δt

{f,H

}+ ...

The same is true for infinitesimal space transformation so that q → q+ δq and so f(q)→ f(q+ δq)expands to:

f(q + δq) ≈ f(q) + δqδf(q)

δq+ ... = f(q) + δq

{f, p}

+ ...

5.2 Finite Transformations

A finite change can be defined by a series of small changes.

limN→∞

(1 +

θ

N

)N= eθ

5.3 Group Theory

δU = εo{U,G}

If ε = δα then in the limit where δα→ 0:

δU

δα={U,G

}This holds for all scalar functions such that if µ =

{U,G

}then:

δµ

δα={µ,G

}={{U,G

}, G} =

δ2U

δα2

This can carry on for more and more scalar functions such that more Poisson’s Brackets are nested.Taylors Theorem is:

U(α) = U

∣∣∣∣α=0

+ αδU

δα

∣∣∣∣α=0

+1

2α2 δU

2

δα2

∣∣∣∣α=0

+ ...

In terms of Poisson’s Brackets this can be written:

U(α) = U

∣∣∣∣α=0

+ α{U,G

}∣∣∣∣α=0

+1

2α2{{U,G

}, G}∣∣∣∣α=0

+ ...

This can be related to the Taylor expansion of ex = 1 + x+ 12x

2 + ....

12

5.4 Operators

If an operator G ={·, G}

is introduced then G · U ={U,G

}such that:

U(α) = eαGU

∣∣∣∣α=0

=

(1 + αG+

1

2α2G · G+ ...

)U

∣∣∣∣α=0

U is the element of the group and G is the element of the algebra. In quantum mechanics, thegenerator of a finite change in time from t→ to is the Hamiltonian:

|ψ(t)〉 = e−ih (t−to)H|ψ(to)〉

5.5 Transformations

5.5.1 Translation

As the momentum is the generator of space G→ p ={·, p}

. The translation can then be writtenas:

eαpU(xo) = U

∣∣∣∣xo

+ α{U, p

}∣∣∣∣xo

+1

2α2{{U, p

}, p}∣∣∣∣xo

+ ...

As{U, p

}= ∂U

∂x then Taylor’s Theorem for U(xo + a) can be found:

eαpU(xo) = U

∣∣∣∣xo

+ α∂U

∂x

∣∣∣∣xo

+1

2α2 ∂

2U

∂x2

∣∣∣∣xo

+ ...

5.5.2 Rotation

The angular momentum is the generator for rotations. In the xy-plane G→ Lz ={·, Lz

}.

x(ψ) = x

∣∣∣∣0

+ ψ{x, Lz

}∣∣∣∣0

+1

2ψ2{{x, Lz

}, Lz

}∣∣∣∣0

+ ...

In the x direction{x, Lz

}= −

{x, yPx

}= −y and in the y direction

{y, Lz

}={y, xPy

}= x.

x(ψ) =

[1− 1

2ψ2 + ...

]x

∣∣∣∣0

− [ψ + ...] y

∣∣∣∣0

= xo cosψ − yo sinψ

This can be calculated for y(ψ) in the same way and gives y(ψ) = xo sinψ + yo cosψ

5.6 Combining Finite Transformations

Translations can be calculated by the product of two generators, eaAebBU = eaA+bBU , but ingeneral this does not work for generators.

eaAebBU =

(1+ aA+

1

2a2A · A+ ...

)(1+ bB +

1

2b2B · B + ...

)U

By expanding the brackets and completing the square on the terms:

eaAebBU =

(1+ aA+ bB +

1

2

(aA+ bB

)2+

1

2ab[A, B

]+ ...

)U

Which means that the combination of transformation gives eaAebBU = e(aA+bB+ 12ab[A,B]+...)U .

Finite rotations are non-commutative and so, if there is a rotation in x and then in y, the overallrotation is a rotation about a diagonal. This relation is the Baher-Campbell-Hausdorff relation:

[A, B]U = ABU − BAU = A{U,B

}− B

{U,A

}=

{{U,B

}, A

}−{{

U,A}, B

}

13

Using the Jacobi Identity:{A,{B,U

}}+

{B,{U,A

}}= −

{U,{A,B

}}If{A,B

}= C then the Baher-Campbell-Hausdorff relation becomes:

[A, B]U ={U,C

}= C

5.7 Newtonian Symmetries

The Poisson Brackets of the momentum, angular momentum and Galilean boost have certainvalues.

5.7.1 Momentum{Pi, Pj

}= 0

{Li, Pj

}= −εijkPk

{Bi, Pj

}= −mδij

5.7.2 Angular Momentum{Pi, Lj

}= εijkPk

{Li, Lj

}= εijkLk

{Bi, Lj

}= εijkBk

5.7.3 Galilean Boost{Pi, Bj

}= mδij

{Li, Bj

}= −εijkBk

{Bi, Bj

}= 0

5.7.4 Space-time

In space-time coordinates, rotations and boosts combine to give space-time rotations.

14

6

Statistical Mechanics

6.1 Canonical Ensemble

A canonical ensemble is system of N particles with a fixed volume V , which is in contact with athermal bath with a temperature T .

6.2 Statistical Identities

6.2.1 Helmholtz Free Energy

The mean energy is U(V, T ) and the entropy is S(V, T ). Using these the Helmholtz free energycan be defined.

F (V, T ) = U − TS

6.2.2 Partition Function

Z(V, T ) =∑α

e−βEα = e−βF

Where β = 1kBT

. This means that the Helmholtz free energy can be written as:

F = −kBT lnZ

This links the microstates with the macrostates.

6.2.3 Gibbs Law

Gibbs Law gives the relation of the Entropy to the probability of microstates.

S = −kB∑α

Pα lnPα

6.3 Probability

To obtain the probability the mean energy U =∑α PαEα and Gibb’s Law are placed into the

Helmholtz Free Energy. As the system is constrained to∑α Pα = 1 then a Lagrange multiplier is

also added.F =

∑α

PαEα + kBTPα lnPα + λ∑α

Pα

Minimising the free energy determines the values of the variable. The Euler-Lagrange equationsfor this system are:

∂F

∂Pα+∂F

∂λ= 0

15

And so the probability is constant as ∂F∂Pα = Eα+λ+kBT [lnPα + 1] = 0. This gives the probability

as:Pα = e−βEαe−1+βλ

Calculating λ gives Z−1 and so the probability is:

Pα =e−βEα

Z

6.4 Phase Transistions

Within the Free energy the mean energy tends to favour order and so it dominates at low tem-peratures. This means that the system becomes more ordered as the temperature decreases. Theentropy tends to favour disorder and so it dominates at high temperatures. The change fromdisorder to order can be smooth or sudden. Sudden changes are phase transitions.

6.5 Magnetism

The change of a paramagnet to a ferromagnet can be used to study the statistical mechanics of asystem.

Figure 6.1: Magnetic Lattice

A lattice with sites labelled i, j, k, etc. have elements of magnetic moment place on them. These spinsites can either be positive or negative σi = ±1. A microstate is labelled σ˜ = (σ1, σ2, σ3, ..., σN ).

If there are N sites thant the number of microstates is N = 2N .{α}

={

1, 2, 3, ..., 2N}

If there are 4 sites then there are 16 microstates. The macroscopic properties are given by Mα =∑Ni=1 σ

αi . The energy is calculated using the Ising model such that each spin state is dependent

on its nearest neighbour.

E(σ˜) = −J∑〈i,j〉

σiσj

The system has cyclic period boundaries so that the fourth state neighbours the first.

1

2

34

Figure 6.2: State 4 is connected to state 1

16

If both of the spins are in the same direction then E = −J and if they are in opposite directionsthen E = J . This states that the as the energy decreases the neighbouring spins are alignedirrespective of orientation. There energy is also up-down symmetrical such that if all the spins areflipped then the energy does not change.

Table 6.1: Microstates of 4 Spin States

Microstate Spin Energy

- - - - 4 -4J+ - - -

−2 0- + - -- - + -- - - ++ + - -

0

0+ - + - 4J+ - - + 0- + + - 0- + - + 4J- - + + 0+ + + -

2 0+ + - ++ + - ++ - + +- + + ++ + + + 4 -4J

6.5.1 Low Temperature Probability

If the probability is given by P(σ˜) =∏i δσi,±1 then the only possibility is the microstate where

σ˜ = (1, 1, ..., 1) or σ˜ = (−1,−1, ...,−1). This has S = 0 and so the mean energy, U , dominates thefree energy, F . This means that the free energy is equal to the minimum energy of the system.This form of probability only exist when T = 0K.

6.5.2 High Temperature Probability

If the probability is given by P =∏i12 (δσi,1 +δσi,−1) such that each spin has equal probability then

all possible microstates have equal probability. This maximises the entropy, S = kBN ln 2 � 0.This means that the entropy dominates the free energy as U � TS. This is a good model whenT →∞.

6.5.3 Other Temperature Probability

For other temperatures the probability is P =∏i

(1+m2 δσi,1 + 1−m

2 δσi,−1

)where m is the factor

used to determine the probability at temperatures not at 0 or∞. The magnetisation of the systemis given by M = 〈

∑i σi〉 = Nm. The mean energy is U =

⟨E(σ˜)

⟩= −J

∑〈i,j〉 〈σiσj〉 = −JzNm2

and the entropy is S = −kB((

1+m2

)ln(1+m2

)+(1−m2

)ln(1−m2

)). By minimising the free energy

with respect to m then a self consistent relation can be found.

m = tanh(2βJzm)

This shows that for high temperatures there is only one microstate with non-zero magnetisation,but lower than the critical temperature TC = 2Jz

kBthere is a phase transition from paramagnet

to ferromagnet. The paramagnet has up-down symmetry where M = 0, whereas the ferromagnethas M 6= 0 and so the symmetry is broken. At the phase transition the up-down symmetry isspontaneously broken.

17

m2

TC

Ferromagnet Paramagnet

T

Figure 6.3: Phase Transition

6.6 Dimensions

In different systems each site can have different numbers of neighbours, z.

6.6.1 1D

Figure 6.4: z = 2

The value of m can be analytical solved for both a 1D and 2D system.

6.6.2 2D

(a) z = 4 (b) z = 6

18

6.6.3 3D

Figure 6.5: z=6

m cannot be found analytically for a 3D system. This means only approximations can be madefor m in almost all physical systems.

19

7

Relativity and Space-Time

Space time has four coordinates xµ = (ct, x, y, z) which can be denoted:

xµ = (x0, x1, x2, x3)

7.0.4 Metric

The metric is the space-time version of the Krnecker Delta:

ηµν =

−1 0 0 00 1 0 00 0 1 00 0 0 1

The metric is antisymmetric and so ηµν = ηTµν . This can be shown by ηµσησν = δµν , which inmatrix notation is η−1η = 1.

7.1 The Invariant Interval

The invariant interval remains constant in all reference frames.

δS2 =∑µν

ηµνδxµδxν

As each other indices appears twice in the same equation the summation can be implied by Ein-stein’s Notation.

δS2 = ηµνδxµδxν

7.2 Lorentz Transformations

The transformations from one reference frame to another is δxµ′ = Λµν δxµ, which can be seen from

the invariant interval ηµνδxµ′δxν ′ = ηµνδx

µδxν . The repeated symbols can be replaced to giveηµνΛµσΛν% = ησ%. In matrix notation ΛT ηΛ = η.

Λµν =

γ −βγ 0 0−βγ γ 0 0

0 0 1 00 0 0 1

Where β = v

c and γ = 1√1−β2

. The transformation for a boost is xµ′ = Λµνxν .

20

ct′

x′

y′

z′

=

γ(ct− βx)γ(x− βct)

yz

21

8

Electromagnetism

Maxwell’s equations for electromagnetism are space-time invariant. The conservation law for elec-tromagnetism ∂%

∂t + ∇ · j˜ = 0 only has three components, but the j0 component can be found

using:∂0(c%) + ∂ij

i = 0

Where ∂µ = ∂∂xµ

. If j0 = c% then the conservation law can be written in the invariant form as:

∂µjµ = 0

8.1 Electromagentic Fields

The electric and magnetic fields, E˜ and B˜ , cannot extend to four vectors and so the scalar andvector potentials must be used.

E˜ = −∂A˜∂t−∇φ

B˜ = ∇×A˜The invariant form of these equations is the field strength tensor:

Fµν = ηµσ∂σAν − ην%∂%Aµ

=

0 E1

cE2

cE3

c

−E1

c 0 B3 B2

−E2

c −B3 0 B1

−E3

c B2 −B1 0

8.2 Sourced Maxwell Equations

8.2.1 Gauss’ Law for Electric Fields

Gauss’ law for electric fields is ∇·E˜ = %εo

. This can be made invariant by taking the index version1c∂iE

i = cεo

%c2 , and as the magnetic permeability can be written εoc

2 = µ this gives −∂iF i0 = µoj0.

As F 00 = 0 this can be extended to:−∂µFµ0 = µoj

0

8.2.2 Ampre’s Law

Ampre’s Law states that ∇ × B˜ − µoεo ∂E∂t = µoj˜, which in terms of the field strength tensor is

∂µFµi = µoj

i because F ij = −F ji. Again, as F 00 = 0 the wholly invariant equation is:

∂µFµν = −µojµ

22

8.3 Maxwell Equations for Electromagnetism

8.3.1 Field Strength Tensor

Fµν = ηµσ∂σAν − ην%∂%Aµ

8.3.2 Gauss’ Law

∂µjµ = 0

8.3.3 Ampre’s Law

∂µFµν = −µojν

8.4 Transformations

The transformations of the electric and magnetic fields can be found from xµ′ = Λµσxσ and so the

transformation of the field strength tensor is:

Fµν ′ = ΛµνΛν%Fσ%

23

9

Relativistic Actions

If the action is to be relativistic then a unique action can be defined:

S = −mc2∫δτ

As proper time is τ =√

1− v2

c2 t the action can be written S = −mc2∫δt√

1− v2

c2 , which in the

non-relativistic limit |v| � c

S ≈ −mc2∫δt(1− v2

2c2+ ...) =

∫δt

1

2mv2 + ...

Which agrees with the Newtonian mechanical kinetic energy T = 12mv

2. In the relativistic limitwhere v → c:

S = −mc∫δτ√−ηµνUµUν

Where U2 = ηµνUµUν =

ηµνδxµδxν

δτ2 = −c2. The Euler-Lagrange equations can be applied to theaction to obtain the equations of motion.

L = −mc√−ηµνUµUν

xµ = ∂Tµ∂τ = Uµ so the Euler-Lagrange equations are:

δ

δt

∂L∂Uµ

− ∂L∂xµ

= 0

As xµ is cyclic then ∂L∂Uµ must be constant, and as ∂L

∂Uµ = cUµ then the velocity of the particlemust be constant.

δ

δτ

δxµ

δτ= 0

9.1 Relativistic Forces

In a Newtonian system the force is given by F˜ = δpδt = m

δv

δt and so in the relativistic case the timechanges from t→ τ , the velocity changes from v → Uµ and the momentum from p→ mUµ.

Fµ = mδUµ

δτ

As δδτ (−c2) = 0 it can be written as δ

δτ (ηµνUµUν) = 0 which leads to:

ηµνδUµ

δτUν + ηµνU

µ δUν

δτ= 0

24

ηµνUµ δUν

δτ = 0 and so ηµνUµFν = 0, which is the analogue of showing v˜ · F˜ = 0. The zero

component of the relativistic force can be found by using:

U0F0 =∑i

U iF i

And because U0 = c δtδτ = γc and U i = δxi

δτ = γvi them the zero component of the force is:

F0 =1

cv˜ · F˜

This has the dimensions of the power input into the particle. For an electromagnetic system therelativistic force can be found. The Lorentz force of classical mechanics is F˜ = q

(E˜ + v˜×B˜),

which, in the z direction is δp3

δt = q(E3 + v1B2 − v2B1). The relativistic form in terms of the fieldstrength tensor is:

1

γ

δp3

δτ=

1

γq(−η00U0F 03 − η11U1F 13 − eta22U2F 23 − η33U3F 33)

η33U3F 33 can be added arbitrarily as F 33 = 0. It is apparent that there is a summation in all

three direction to give:δpi

δτ= −qηµνUµF νi

This is not manifestly relativistically covariant as the indices do not include the zero component.As the zero component of the force is the rate of change of energy then δε

δt = qv˜ ·E˜ where ε = cp0.

This means that ∂p0

∂τ = −qηµνUµF ν0 and so the explicit realisation of the electromagnetic force is:

δpσ

δτ= −qηµνUµF νσ

25