2 Space-time transformations

Quantum Field Theory-I Prof. G. Isidori

UZH and ETH, HS-2017 Revised version: October 15, 2017

http://www.physik.uzh.ch/en/teaching/PHY551/HS2017.html

2 Space-time transformations

2.1 Relativistic notations

In most of this course we adopt the so-called system of natural units, namely we set

¯

h = c = 1 (1)

In this system of all physical quantities can be epxressed in units of energy (by means of an implicit insertion of appropriate powers of ¯ h and c): [length] = [time] = [energy]

⁻¹

= [mass]

⁻¹

.

Our convention for the metric tensor is

g

_µν

= diag(1, −1, −1, −1) = g

^µν

(2) The four-vector for space-time position is

x

^µ

= (t, ~ x) = (x

⁰

, ~ x) x

_µ

x

^µ

= x

_µ

g

^µν

x

_ν

≡ x

²

= t

²

− |~ x|

²

(3) and the four-momentum is

p

^µ

= (E, ~ p) = (p

⁰

, ~ p) p

_µ

p

^µ

= p

_µ

g

^µν

p

_ν

≡ p

²

= E

²

− |~ p|

²

(4) Lorentz transformations are rotations and boosts in space-time, generically denoted by

x

^µ

→ (x

⁰

)

^µ

= Λ

^µ_ν

x

^ν

(5)

By construction, Lorentz transformations leave invariant scalar products x

²

= (x

⁰

)

²

←→ t

²

− |~ x|

²

| {z }

invariant under Lorentz transf.

(6)

Translations are coordinate transformations of the type

x

^µ

→ (x

⁰

)

^µ

= x

^µ

− a

^µ

(7)

where a

^µ

does not depend on x

^µ

. Finally, we denote as Poincar´ e transformations the combined effect of Lorentz transformations and translations

x

^µ

→ (x

⁰

)

^µ

= Λ

^µ_ν

x

^ν

− a

^µ

(8) Throughout this course we will consider theories which are invariant under these coordinate transformations. As we discuss in this lecture, these transformations for a group. This property has important implications in finding conserved quantities, and in classifying the allowed forms of fields and actions.

2.2 Groups and representations

A set of elements G = {g

₁

, g

₂

, . . .} is said to form a group if we can define a multi- plication operation, g

i

· g

j

, which satisfies the following conditions:

1. g

_i

· g

_j

is still an element of G 2. (g

_i

· g

_j

) · g

_k

= g

_i

· (g

_j

· g

_k

)

3. There exists an element e (the unit element), such that g

i

· e = e · g

i

(for any g

_i

)

4. For any g

_i

we can identify a corresponding element g

_i⁻¹

(the inverse of g

_i

), such that g

i

· g

_i⁻¹

= g

⁻¹_i

· g

i

= e

The abstract and general concept of a group can be made more concrete introducing its representations on a vector space V. If we can associate to each element of the group G a linear operator, T (g

i

), acting on V [T : V → V ], we say that these operators form a representation of the group if

T (g

_i

)T (g

_j

) = T (g

_i

· g

_j

) and T (e) = 1 (9) The representation of the group is nothing but an explicit realization of the group, where the abstract elements g

_i

are replaced by the linear operators T (g

_i

) (e.g. n × n matrices in a n-dimensional vector space) and the multiplication is identified with the ordinary product of operators. The representation is said to be faithful if the relation g

_i

→ T (g

_i

) is one-to-one. If V can be decomposed into at least two subsets (V = V

₁

⊕V

₂

) which are separately left invariant by the linear operators [T : V

₁

→ V

₁

and T : V

2

→ V

2

], then the representation is said to be reducible.

In physics the notion of group is often very useful to describe the transformation

of physical quantities under the symmetries of nature. For example, the rotations on

the plane around a common origin form a group, denoted by SO(2). A representation of this group is given by the 2 × 2 matrices of the type

R(α) =

cos α sin α

− sin α cos α

(10) which acts on the 2-dimensional vector space of the Cartesian coordinates. General- izing this simple case, we can define the group SO(n) of rotations on a n-dimensional Euclidean space.

As can be seen from the example of the rotations, we often deal with groups where the elements can be unambiguously identified by a set of real parameters {α

₁

, . . . α

_n

} (e.g. the rotation angles), which can be varied continuously. Since the product g(α

_i

) · g(β

_i

) must still be an element of the group and, in these cases, it must be expressed in terms of the continuous parameters, we can write

g(α

_i

) · g(β

_i

) = g(γ

_i

) (11) This relation let us to define n functions of the type

γ

_i

= f

_i

(α

₁

, . . . α

_n

; β

₁

, . . . β

_n

) (12) In the case of the rotations, and in many other physical applications, the f

_i

are dif- ferentiable functions of the parameters. Groups of this kind are denoted continuous groups or Lie groups.

The number of real parameters {α

₁

, . . . α

_n

} which characterize a continuous group is called the dimension of the group. By convention, the unit element is associated to the case where all the parameter vanish. A very interesting prop- erty of Lie groups is the fact that their structure is essentially determined by the small perturbations around the unit element. Given a representation T (α

_i

) of G, expanding in powers of the α

_i

, we can write

T (α

i

) = 1 +

n

X

i=1

α

i

X

i

+ O(α

²

) (13)

where the X

_i

are some α-independent linear operators. These are called infinitesimal operators or generators of the representation T (α

_i

) of G. It can be shown that:

I. If two representations of G have the same generators they are the same repre-

sentation.

II. For any representation of G, the generators must satisfy a commutation rela- tion of the type

[X

i

, X

j

] = C

_ij^k

X

k

(14) where the C

_ij^k

, denoted structure constants, are independent of the represen- tation.

III. Any set of operators X

_i

which satisfy the relation (14) provide a set of gener- ators for a representation of G.

A representation is called unitary if its elements obey the relation T

^†

T = 1, or T

⁻¹

= T

^†

. Give the α

_i

are real, it follows that the generators of a unitary representation are anti-Hermitian: X

_i^†

= −X

_i

.

2.2.1 The SO(3) group

The generators of SO(3), or the group of rotations in 3 Euclidean dimensions, satisfy the commutation relations

[L

_i

, L

_j

] = ε

_ijk

L

_k

. (15)

Starting from the L

i

we can define the (Hermitian) operators J

i

= iL

i

, that satisfy the commutation relations

[J

_i

, J

_j

] = iε

_ijk

J

_k

(16)

Moreover, if we combine J

_x

and J

_y

to form J

±

= J

_x

± iJ

_y

, it follows that

[J

_z

, J

±

] = ±J

±

(17)

Denoting by T

^(j)

a generic irreducible representation of SO(3), i.e. using the index j to label the representation, let’s consider within this representation a generic eigenvector of J

z

, with eigenvalue j

z

:

J

_z

|ψ(j, j

_z

)i = j

_z

|ψ(j, j

_z

)i (18) From (17) it follows that (show it):

J

±

|ψ(j, j

_z

)i ∝ |ψ(j, j

_z

± 1)i (19) If the representation is finite-dimensional, there must be a maximum value of j

_z

, which we can identify with the label j of the representation, such that

J

₊

|ψ(j, j )i = 0 (20)

Because of (19), acting repeatedly on |ψ(j, j )i with J

−

one can form a series of n + 1 eigenvectors of J

_z

|ψ(j, j − 1)i = N

j−1

J

−

|ψ(j, j )i,

|ψ(j, j − 2)i = N

j−2

J

−

|ψ(j, j − 1)i,

.. . (21)

up to

J

−

|ψ(j, j − n)i = 0 (22) which again must hold because of the finiteness of the representation.

We can now use the fact that J

²

= P

i

J

_i²

= −X

²

commutes with all the gen- erators and therefore must yield the same eigenvalue for all the eigenvectors of this basis. Using the explicit expression of J

²

in terms of J

_±

and J

_z

, and applying it to Eqs. (20)–(22), one finds (show it):

J

²

|ψ(j, j )i = j(j + 1)|ψ(j, j )i

J

²

|ψ(j, j − n)i = (j − n)(j − n − 1)|ψ(j, j − n)i (23) Since the two eigenvalues must coincide, it follows that

j = n

2 (24)

where n is a positive integer number.

We thus conclude that the irreducible representations of SO(3) can be classified according to the eigenvalues of X

²

on the corresponding vector space. These eigen- values can in turn be labelled by a semi-integer number j = 0,

¹₂

, 1, . . . and have multiplicity 2j + 1 (which corresponds to the dimension of the representation).

Operators such as X

²

, which act like a multiple of the unit operator inside a given representation but distinguish different representations, are called Casimir opera- tors. In every Lie group it is possible to identify a set of Casimir operators and to classify all the irreducible representations in terms of their eigenvalues.

2.3 The Lorentz group

The Lorentz group is defined as the set of transformations over a four-dimensional vector space, with elements x

^µ

, which preserve the scalar product

xy = x

^µ

g

_µν

x

^ν

(25)

Here g

_µν

= g

^µν

is the metric tensor, defined by the matrix

g =









1 0 0 0

0 −1 0 0

0 0 −1 0

0 0 0 −1









(26)

A representation of these transformations is given by the set of 4 × 4 matrices Λ

^µ_ν

x

^µ

→ x

^0µ

= Λ

^µ_ν

x

^ν

(27)

which satisfy the condition

Λ

^µ_λ

Λ

^ν_ρ

g

_µν

= g

_λρ

(28)

Note that, introducing the transpose matrix (Λ

^T

)

_λ^µ

= Λ

^µ_λ

, the condition (28) can be written as Λ

^T

gΛ = g in terms of the ordinary matrix multiplication. From this we deduce that Λ

⁻¹

= g Λ

^T

g.

By convention, we denote by x

^µ

the 4-vector of space-time coordinates (ct, x, y, z) and by x

_µ

= g

_µν

x

^ν

the combination (ct, −x, −y, −z). Any 4-vector transforming as x

^µ

in Eq. (28) is called contravariant vector and is indicated with an upper index (i.e. V

^µ

). On the other hand, any 4-vector transforming as x

_µ

is called covariant vector. As suggested by these names, these two types of vectors have opposite transformation properties:

x

_µ

→ g

_µρ

x

^0ρ

= g

_µρ

Λ

^ρ_σ

x

^σ

= (gΛggx)

_µ

= [(Λ

⁻¹

)

^T

gx]

_µ

= (Λ

⁻¹

)

^ρ_µ

g

_ρλ

x

^λ

= x

_ρ

(Λ

⁻¹

)

^ρ_µ

(29) It can be shown (→ Exercise set n. 3) that

I. The transformations defined by Eqs. (27)–(28) form a group.

II. The entire Lorentz group is not continuous. Only the transformations with det(Λ) = 1 and (Λ

⁰₀

) ≥ 1 form a continuous subgroup denoted proper Lorentz group.

2.3.1 Representations of the Lorentz group

The two basic building blocks of the proper Lorentz group are the boosts along one direction and the rotations around one axis. The boost with velocity β = v/c along the x axis leads to the transformations

ct → ct

⁰

= γ(ct) − γβx

x → x

⁰

= γx − γβ(ct) γ = (1 − β

²

)

^−1/2

(30)

or

Λ(β)

^boost−x

=









γ −γβ 0 0

−γβ γ 0 0

0 0 1 0

0 0 0 1









=









1 −β 0 0

−β 1 0 0

0 0 1 0

0 0 0 1









+ O(β

²

) (31)

while the rotations are completely analog to the case of SO(3).

Combing a generic infinitesimal boost with a generic infinitesimal rotation we obtain the most general infinitesimal proper Lorentz transformation. Writing

Λ

^µ_ν

≈ δ

^µ_ν

+ ω

^µ_ν

(32)

we have

ω =









0 −β

₁

−β

₂

−β

₃

−β

1

0 −α

3

α

2

−β

₂

α

₃

0 −α

₁

−β

₃

−α

₂

α

₁

0









= X

i

(α

i

X

i

+ β

i

Y

i

) (33)

and the six matrices X

i

and Y

i

can be chosen as generators of the proper Lorentz group in this 4 × 4 matrix representation. [The proof that this is the most general infinitesimal transformation follows from the fact that Eq. (28) implies that ω

_σν

= g

σµ

ω

^µ_ν

is a generic antisymmetric matrix].

A generic element of the entire (non-continuous) group can be obtained by com- bining an element of the proper group with the discrete transformations of parity (P ) and time-reversal (T ). Within the 4 × 4 matrix representation:

P =









1 0 0 0

0 −1 0 0

0 0 −1 0

0 0 0 −1









T =









−1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1









P T = −1 (34)

It can be show (→ Exercise set n. 3) that I. The matrices

A

i

= 1

2 (X

i

+ iY

i

) B

i

= 1

2 (X

i

− iY

i

) (35) satisfy the following commutation relations:

[A

_i

, A

_j

] = ε

_ijk

A

_k

[B

_i

, B

_j

] = ε

_ijk

B

_k

[A

_i

, B

_j

] = 0 (36) hence the irreducible representations of the proper Lorentz group can be classi- fied in terms of two semi-integer numbers, j and j

⁰

, which labels the eigenvalues of the two Casimir operators: A

²

= P

i

A

²_i

and B

²

= P

i

B

_i²

.

II. The 4 × 4 representation of the Λ matrices corresponds to the (j, j

⁰

) = (

¹₂

,

¹₂

) representation.

2.4 The Poincar´ e group

The most general class of space-time transformations which connect two inertial reference frames, within special relativity, is the combination of Lorentz “rotations”

(which leave the origin of the coordinates unchanged) and the space-time transla- tions. The combination of these two groups is called inhomogeneous Lorentz group or Poincar´ e group.

A generic element of this group is unambiguously identified by a pair formed by a Lorentz matrix and a constant vector: U(Λ, a). The effect on the coordinates is a Lorentz “rotation” followed by a translation of the origin by a

^µ

x

^µ

→ x

^0µ

= Λ

^µ_ν

x

^ν

− a

^µ

U (Λ, a) = U (1, a)U (Λ, 0) (37) The subgroup of translations is an Abelian group and indeed in this case the

“multiplication operation” of the group leads to a sum of the corresponding param- eters:

U (1, a

^µ

)U(1, b

^µ

) = U (1, a

^µ

+ b

^µ

) (38) This implies that if we define the generators P

^µ

= g

^µν

P

_ν

for the infinitesimal trans- lations

U (1, ) = 1 − i

^µ

P

_µ

+ . . . (39)

they satisfy the commutation relation

[P

^µ

, P

^ν

] = 0 (40)

and the finite transformations can be expressed as U (1, a) = lim

N→∞

1 − i a

^µ

N P

_µ

N

= exp (−ia

^µ

P

_µ

) (41) The situation becomes more involved once we include also Lorentz transforma- tions. In the general case, the generators are conventionally defined as

U(1 + ω, ) = 1 − i

2 ω

_µν

J

^µν

− i

_µ

P

^µ

+ . . . (42) The composition law of two generic Poincar´ e transformations is

U (Λ

₂

, a

₂

)U(Λ

₁

, a

₁

) = U (Λ

₂

Λ

₁

, Λ

₂

a

₁

+ a

₂

) (43)

Using this composition law, it follows that

U

⁻¹

(Λ, a)U (1 + ω, )U (Λ, a) = U (1 + Λ

⁻¹

ωΛ, Λ

⁻¹

+ Λ

⁻¹

ωa) (44) Expanding this result to first order in ω and , and setting a = 0, one gets

U

⁻¹

(Λ, 0)J

^ρσ

U(Λ, 0) = Λ

^ρ_µ

Λ

^σ_ν

J

^µν

(45) U

⁻¹

(Λ, 0)P

^ρ

U(Λ, 0) = Λ

^ρ_µ

P

^µ

(46) These equations imply that J

^µν

and P

^µ

transform as a contravariant tensor and vector, respectively (see next subsection).

Treating Λ and a as small parameters and expanding Eq. (44), keeping also the mixed terms, one can derive the following commutation relations

[J

^µν

, J

^ρσ

] = i (g

^νρ

J

^µσ

− g

^µρ

J

^νσ

− g

^νσ

J

^µρ

+ g

^µσ

J

^νρ

) (47) J

^µν

, P

^λ

= i g

^νλ

P

^µ

− g

^µλ

P

^ν

(48) Together with Eq. (40), these results determine the structure of the Poincar´ e group.

It is easy to verify that P

²

= P

_µ

P

^µ

commutes with all the generators of the group.

Given the special role of the Hamiltonian operator H = P

⁰

, it is worth to derive its commutation relations with the other generators. Renaming them as follows (i.e.

grouping them into three 3-vectors):

P = {P

¹

, P

²

, P

³

} J = {J

¹⁰

, J

²⁰

, J

³⁰

} K = {J

²³

, J

³¹

, J

¹²

} (49) the following identities holds:

[J

_i

, H] = [P

_i

, H] = 0 [K

_i

, P

_j

] = iHδ

_ij

(50) [K

i

, H] = iP

i

[J

i

, P

j

] = iε

ijk

P

k

(51) 2.4.1 Transformations of fields and observables

We are now ready to discuss the effects of Poincar´ e transformations on a field Ψ

_r

(x) of the space-time coordinates or, more in general, of any local observable O(x) acting on the Hilbert space. In general the effect is a twofold action: a change of the coordinates and a linear transformation of the field components (here denoted by the generic subscript r). In order to distinguish these two effects, let’s first consider the effect of a pure Lorentz transformation. In this case we can write

Ψ

⁰_r

(x

⁰

) = S

_rs

(Λ)Ψ

_s

(x) = S

_rs

(Λ)Ψ

_s

(Λ

⁻¹

x

⁰

) (52)

where S

_rs

(Λ) is a generic representation of the Lorentz group. The effect on the co- ordinates, which may appear odd at first sight, can easily be understand choosing as particular example the case {Ψ

s

(x)} = {x

µ

}. The classification in irreducible repre- sentations of the Lorentz group provides, in general, a very useful tool to distinguish the various types of fields and operators.

In the first part of this course we have already encountered scalar fields and operators (such as the Lagrangian). These belongs to the (0, 0) representation of the Lorentz group, for which Eq. (52) reduces to

φ(x) = φ(Λ

⁻¹

x) (53)

In the previous section of this note we have also seen examples of vector fields, i.e. of (

¹₂

,

¹₂

) representations, for which Eq. (52) reduces to

V

^µ0

(x) = Λ

^µ_ν

V

^ν

(Λ

⁻¹

x) (54) and of antisymmetric tensors belongings to the (1, 0) and (0, 1) representations. In the next set of lectures we shall introduce appropriate spinor fields transforming as (

¹₂

, 0) and (0,

¹₂

).

If we now consider the field (or the observable) as an operator in the Hilbert space, thanks to the Wigner theorem we can identify Ψ

⁰_r

(x

⁰

) with the unitary trans- formation U (Λ, 0)

^†

Ψ

_r

(x

⁰

)U (Λ, 0). Taking into account Eq. (52) this implies

U(Λ, 0)

^†

Ψ

_r

(x)U (Λ, 0) = S

_rs

(Λ)Ψ

_s

(Λ

⁻¹

x) (55) U(Λ, 0)Ψ

_r

(x)U (Λ, 0)

^†

= S

_rs

(Λ

⁻¹

)Ψ

_s

(Λx) (56) Generalizing this result to the case of generic Poincar´ e transformations, we finally obtain

U (Λ, a)

^†

Ψ

_r

(x)U (Λ, a) = S

_rs

(Λ)Ψ

_s

(Λ

⁻¹

(x + a)) (57)

U (Λ, a)Ψ

_r

(x)U (Λ, a)

^†

= S

_rs

(Λ

⁻¹

)Ψ

_s

(Λx − a) (58)