fermions

matter

spin 1/2

leptons

baryons

 

 

bosons

forces

spin 1 (graviton spin 2)

interactions vector bosons

QED

WEAK

QCD

Gravity G

 

Higgs Boson

Standard model of elementary particles. The electron is at lower left.

 

Quark flavor properties[67]

Name

Symbol

Mass (MeV/c2)*

J

B

Q

I3

C

S

T

B′

Antiparticle

Antiparticle symbol

First generation

Up

u

1.7 to 3.3

12

+13

+23

+12

0

0

0

0

Antiup

Down

d

4.1 to 5.8

12

+13

13

12

0

0

0

0

Antidown

Second generation

Charm

c

1,270+70−90

12

+13

+23

0

+1

0

0

0

Anticharm

Strange

s

101+29−21

12

+13

13

0

0

−1

0

0

Antistrange

Third generation

Top

t

172,000900  1,300 

12

+13

+23

0

0

0

+1

0

Antitop

Bottom

b

4,190+180−60

12

+13

13

0

0

0

0

−1

Antibottom

 

J = total angular momentum, B = baryon number, Q = electric charge, I3 = isospin, C = charm, S = strangeness, T = topness, B′ = bottomness.
* Notation such as 4,190
+180−60 denotes measurement uncertainty. In the case of the top quark, the first uncertainty is statistical in nature, and the second is systematic.


  • leptons and baryons have antiparticles
  • charged vector bosons have antiparticles
  • , Higgs are considered to be particle & antiparticle
  • gluons are also particle antiparticle but not as simple. There are three colors and three anticolors . The gluon states can carry color (charged). We need 8 gluons and there are 9 unique combinations of the color states . The analysis of the symmetry group SU(3) selects 8 combinations (If U(3) was the symmetry there would be a singlet state. Remember that a spin 1 particle built from 2 spin 1/2 particle can combine to have Sz=0. A color state can combine colors with a total of zero color and still not be a singlet.)

 

color singlet .This state is not included.

Again color singlet and are not the same thing just as Jz=0 can be either J=1 or J=0 with a total spin of 1 still having a z-component of zero so a cancellation of the color charge doesnt imply a singlet state. It is the color singlet state that is stable.

 

  • The details of QCD are typically glossed over in introductory particle physics. The reason is that the forces are so strong that the underlying interactions are simply used to select the allowable quark states. This is how we treat the Feynman diagrams for QCD. Assume color singlets and then use these states to explain the interactions (pion exchange).

The weak interaction is unique in a number of respects:

  1. It is the only interaction capable of changing flavour.
  2. It is the only interaction which violates parity symmetry P (because it only acts on left-handed particles). It is also the only one which violates CP.

It is mediated by heavy gauge bosons. This unusual feature is explained in the Standard Model by the Higgs mechanism.

 

 

 

 

Quantum numbers -------------------------------

 

Quantum numbers are the observables of the quantum system. Under the action of a symmetry some quantum numbers will not change. For example under reflection, P, the location of a particle will change. Under rotation the location of a particle will change. However the even or odd character of a wf doesnt change under reflection and the angular momentum of a state is unchanged by rotation.

Quantum numbers and symmetry (M=almost always conserved, A=always, P=partially)

 

Relevant

Symmetry

 

int.

violate

int.

follow

 

Poincare space-time

Translations, Rotations, Boosts

momentum conservation

ang. mom conservation

energy conservation

 

all

 

Parity

P

Improper space-time

 

W

S,E

P

There is no observed quantum number conserved for time reversal because it is not a unitary transformation.

Charge Conjugation

C

Particle anti particle

 

W

S,E

M

 

 

 

 

 

 

charge

Q

phase U(1)

 

 

all

A

Baryon number

B

phase U(1)

 

 

all

A

lepton number

L

phase U(1)

 

 

all

 

Lepton F

 

phase U(1)

 

mix

S,E

 

Strangeness

S

U(1)

 

W

S,E

P

Isospin (upness, downness)

SU(2)

 

W

S,E

P

Top

 

 

W

S,E

P

Bottom

 

 

W

S,E

P

There are Quantum numbers that are linear combinations of other quantum numbers. These may be more convenient for labeling states. This occurs when the natural quantum states may be linear combinations of states with different quantum numbers. The which has both up and down quarks and is labled by isospin rather Iz rather than .

 

 

 

 

 

 

strong hypercharge

Y=2(Q-Iz)

flavor

(limit as quark mass0)

labels the flavor SU(3) multiplets

W

 

P

Image:Eg2.png

For SU(3) the flavor symmetry of up, down, strange one needs to labels for the multiplets. Usual choice is

weak hypercharge

 

 

 

 

 

 

 

All quarks

just u,d,s

Q

B

Iz

S

 

 

C

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

charge

mass

MeV

I

isospin

J

spin

P

parity

C

charge

conj

lep #

&

LF#

 

 

e

-1

0.5

N/A

1/2

+1

N/A

1,e=1

 

 

0

~0

N/A

1/2

+1

N/A

1,e=1

 

 

-1

106

N/A

1/2

+1

N/A

1,=1

 

 

0

~0

N/A

1/2

+1

N/A

1,=1

 

 

-1

1777

N/A

1/2

+1

N/A

1,=1

 

 

0

~0

N/A

1/2

+1

N/A

1,=1

 

 

u

2/3

2

1/2

1/2

+1

N/A

0

 

 

d

-1/3

4

1/2

1/2

+1

N/A

0

 

 

c

2/3

1300

0

1/2

+1

N/A

0

 

 

s

-1/3

90

0

1/2

+1

N/A

0

 

 

t

2/3

172000

0

1/2

+1

N/A

0

 

 

b

-1/3

4000

0

1/2

+1

N/A

0

 

 

0

0

0,1

1

-1

-1

0

 

 

1

80000

N/A

1

 

N/A

0

 

 

-1

80000

N/A

1

 

N/A

0

 

 

0

90000

N/A

1

 

-1

0

 

 

g

0

0

0

1

-1

 

0

 

 

G

0

 

N/A

2

 

 

0

 

 

H

0

>115000

N/A

0

 

 

0

 

 

 

 

 

 

 

 

 

 

 

 

 

  • assign an overall lepton # +1 leptons, -1 antileptons, 0 , all others 0
  • assign generation specific Q#, e-number, μ-number, τ-number but these are not conserved due to neutrino oscillations.
  • Since neutrinos do have a tiny nonzero mass, neutrino oscillation LF (lepton Flavor) only approximate. This means the conservation laws are violated, although because of the smallness of the neutrino mass they still hold to a very large degree for interactions containing charged leptons. However, the lepton number conservation law must still hold (under the Standard Model).
  • Fermions have parity +1 intrinsic and antifermions have -1.
  • Parity for the photon is chosen to be consistent with the convention that the electric field is a vector and the magnetic field is an axial vector. -1
  • Charge conjugation holds for particles that are their own anti particle and it is
  • Vectors change sign under a parity (reflection) axial vectors dont change sign.
  • Wave functions expressed in spherical harmonics or orbital angular momentum states transform with an overall factor
  • For composite systems the parity will depend on the relative orbital angular moment which for two particles which is straightforward to define . For three particle composite systems the wave function is a bit more complicated but the parity can be determined.
  • Combining the parity for the meson where L is the relative wf angular momentum.
  • To look at C we must have so that the states are their own anti particle.

 

Quark flavor properties[41]

Name  ↓

Symbol  ↓

Gen.  ↓

Mass (MeV/c2)  ↓

I  ↓

J  ↓

Q  ↓

S  ↓

C  ↓

B′  ↓

T  ↓

Antiparticle  ↓

Up

u

1

000002.81.5 to 3.3

1/2

1/2

+2/3

0

0

0

0

Antiup

Down

d

1

000004.83.5 to 6.0

1/2

1/2

−1/3

0

0

0

0

Antidown

Charm

c

2

001270.01,270

0

1/2

+2/3

0

+1

0

0

Anticharm

Strange

s

2

000104.0104

0

1/2

−1/3

−1

0

0

0

Antistrange

Top

t

3

171200.0171,200

0

1/2

+2/3

0

0

0

+1

Antitop

Bottom

b

3

004200.04,200

0

1/2

−1/3

0

0

−1

0

Antibottom

(Key: Gen. = generation, I = isospin, J = spin, Q = electric charge, S = strangeness, C = charm, B′ = bottomness, T = topness. Notation like 104+26−34 denotes measurement uncertainty: the value is between 104 + 26 = 130 and 104 − 34 = 70, with 104 being the most likely value.

 

Bosons

Name

Symbol

Antiparticle

Charge (e)

Spin

Mass (GeV/c2)

Force mediated

Existence

Photon

γ

Self

0

1

0

Electromagnetism

Confirmed

W boson

W

W+

−1

1

80.4

Weak

Confirmed

Z boson

Z

Self?

0

1

91.2

Weak

Confirmed

Gluon

g

Self?

0

1

0

Strong

Confirmed

Graviton

G

Self

0

2

0

Gravity

Unconfirmed

Higgs boson

H0

Self?

0

0

> 112

See below

Unconfirmed

 

Standard model of elementary particles. The electron is at lower left.

The weak interaction is unique in a number of respects:

  1. It is the only interaction capable of changing flavour.
  2. It is the only interaction which violates parity symmetry P (because it only acts on left-handed particles). It is also the only one which violates CP.
  3. It is mediated by heavy gauge bosons. This unusual feature is explained in the Standard Model by the Higgs mechanism.

unification is accomplished under an SU(2) U(1) gauge group

 

Hypercharge and Isospin

 

In looking at quantum numbers for the quarks one defines isospin and hypercharge. The weak interaction sector also will define a version of isospin and hypercharge. The strong interaction versions are related to flavor symmetry while the weak versions are based on gauge transformations that unify E&M and weak. These are related but not the same thing. The weak sector combines states (u,d) where the Cabbibo angle is (or CMK matrix) used to define the quark states as mixtures. Strong isospin has (u,d). Also the weak sector adds the leptons. So we expect that if we define strong isospin and hyper charge (strangeness) there should be a similar extension to these ideas incorporated into the weak interaction but the states of the weak interaction are not the same as the states of the strong interaction.

Also the U(1) interaction gets rotated by the mixing. Start with this U(1) and YW as the generator of this transformation.

The Weinberg angle or weak mixing angle is a parameter in the Weinberg-Salam theory of the electroweak force, and is usually denoted as θW. It is the angle by which spontaneous symmetry breaking rotates the original W0 and B0 vector boson plane, producing as a result the Z0 boson, and the photon.  \begin{pmatrix}
Photon \\
Z^0 \end{pmatrix} = \begin{pmatrix}
\cos \theta_W & \sin \theta_W \\
-\sin \theta_W & \cos \theta_W \end{pmatrix} \begin{pmatrix}
B^0 \\
W^0 \end{pmatrix}

 

 

The following discussion will attempt to clarify the difference.

 

 

Ystrong The hypercharge was added to Iz isospin z-component in order to define a 2-d plane for SU(3) flavor u,d,s. Y (y-axis) Iz (x-axis) so that points labeled with Y, Iz are particles in a SU(3) multiplet.

 

Image:Eg2.pngIf u,d,s quarks are the same under the strong interaction than multiplets of SU(3) are degenerate and these particle states will be important. To characterize the members of the multiplet you use two labels Ystrong, Iz.

 

 

Ystrong= S+B=2(Q- Iz)

 

 

 

 

 

 

 

The symmetry for the electroweak unification combines an SU(2) and U(1) symmetry constraint. The SU(2) structure is apparent because we group many of the particles to doublets and a U(1) symmetry should generate an E&M interaction. For E&M one discovers that the generator is the charge. So one would recognize the U(1) symmetry as the global symmetry that requires charge conservation. Build the electroweak however this particular symmetry reveals itself only after the symmetry breaking mixes the vector bosons. Thus we start with a U(1) with the generator Y, weak hypercharge.

 

Yw is the quantity associated with the U(1) gauge transformation that produces the electroweak interaction

 

SU(2)LU(1)Y

 

The SU(2) doublets are even referred to as weak isospin. These weak isospin doublets are all of the weak partners (u,d), (c,s), (t,b), (e,υ), [Strong isospin is u,d with no Cabbibo mixing]. In developing the SU(2) symmetry the operators T

 

with

 

are used. This is the same structure as the strong isospin with I ()

 

start with SU(2) coupling g

U(1) coupling g

after sym breaking e=gsin θW, e=electric charge

 

However the final version of the theory mixes the state associated with the z-components of gauge boson for SU(2) and the gauge boson of U(1).

 

Q  =  (e/g) T3 + (e/g') Y  =  sinθW T3 + cosθW Y,

where we have introduced the Weinberg angle, θW. In terms of this, one can write

Zμ=cosθW W3μ - sinθW Bμ,   and   Aμ=sinθW W3μ + cosθW Bμ.

Which might lead to a redefintion of of T3 and Y such that (In one ref. I found a statement that there is some rescaling of the def. of Y but that it is not very common. Not sure if this is what is reffered to here.)

 

Iz=sinθW T3 and YW= 2cosθW Y

 

sin2θW = 0.2397 cos2θW= 0.7603 cosθW= 0.872 sinθW = 0.490

 

However most references seem to keep Y as the generator of the original U(1) introduced before the mixing.

In any case this is just multiplication by a constant which would seem to be a small factor that you would just need to keep track of i.e are you rescaling or not? Note that in some derivations the states are chosen to satisfy the relationship below. Ie Y operating produces the eignevalues that match the condition below. The operators in this case would not be rescaled. Normally the coupling g, g are the things that get rescaled above perhaps these couplings are assumed to be directly related to Y and T and there is no g g?

 

 

Q = I^W_z + {1 \over 2} Y^W

 

Parameters of the Standard Model

Symbol

Description

Renormalization
scheme (point)

Value

me

Electron mass

 

511 keV

mμ

Muon mass

 

106 MeV

mτ

Tau lepton mass

 

1.78 GeV

mu

Up quark mass

(\mu_{\overline{\text{MS}}}=2\text{ GeV})

1.9 MeV

md

Down quark mass

(\mu_{\overline{\text{MS}}}=2\text{ GeV})

4.4 MeV

ms

Strange quark mass

(\mu_{\overline{\text{MS}}}=2\text{ GeV})

87 MeV

mc

Charm quark mass

(\mu_{\overline{\text{MS}}}=m_c)

1.32 GeV

mb

Bottom quark mass

(\mu_{\overline{\text{MS}}}=m_b)

4.24 GeV

mt

Top quark mass

(on-shell scheme)

172.7 GeV

θ12

CKM 12-mixing angle

 

0.229

θ23

CKM 23-mixing angle

 

0.042

θ13

CKM 13-mixing angle

 

0.004

δ

CKM CP-Violating Phase

 

0.995

g1

U(1) gauge coupling

(\mu_{\overline{\text{MS}}}=M_\text{Z})

0.357

g2

SU(2) gauge coupling

(\mu_{\overline{\text{MS}}}=M_\text{Z})

0.652

g3

SU(3) gauge coupling

(\mu_{\overline{\text{MS}}}=M_\text{Z})

1.221

θQCD

QCD Vacuum Angle

 

~0

Μ

Higgs quadratic coupling

 

Unknown

Λ

Higgs self-coupling strength

 

Unknown

No neutrino mixing is included here.

 

 

Technically, quantum field theory provides the mathematical framework for the standard model, in which a Lagrangian controls the dynamics and kinematics of the theory. Each kind of particle is described in terms of a dynamical field that pervades space-time. The construction of the standard model proceeds following the modern method of constructing most field theories: by first postulating a set of symmetries of the system, and then by writing down the most general renormalizable Lagrangian from its particle (field) content that observes these symmetries.

The global Poincar symmetry is postulated for all relativistic quantum field theories. It consists of the familiar translational symmetry, rotational symmetry and the inertial reference frame invariance central to the theory of special relativity. The local SU(3)\timesSU(2)\timesU(1) gauge symmetry is an internal symmetry that essentially defines the standard model. Roughly, the three factors of the gauge symmetry give rise to the three fundamental interactions. The fields fall into different representations of the various symmetry groups of the Standard Model (see table). Upon writing the most general Lagrangian, one finds that the dynamics depend on 19 parameters, whose numerical values are established by experiment. The parameters are summarized in the table at right.

[edit] The QCD sector

Main article: Quantum chromodynamics

[edit] The electroweak sector

Main article: Electroweak interaction

The electroweak sector is a Yang-Mills gauge theory with the symmetry group U(1)\times SU(2)_L,

\mathcal{L}_{EW}=\sum_\psi\bar\psi\gamma^\mu\left(
i\partial_\mu-g'{1\over2}Y_WB_\mu
-g{1\over2}\vec\tau_L\vec W_\mu
\right)\psi 
+\mathcal{L}_{YM}(B_\mu)
+\mathcal{L}_{YM}(\vec W_\mu),

where Bμ is the U(1) gauge field; YW is the weak hypercharge the generator of the U(1) group; \vec{W}_\muis the three-component SU(2) gauge field; \vec{\tau}_Lare the Pauli matrices infinitesimal generators of the SU(2) group, the subscript L indicates that they only act on left fermions; g' and g are coupling constants.

Additional Symmetries of the Standard Model

From the theoretical point of view, the standard model exhibits additional global symmetries that were not postulated at the outset of its construction. There are four such symmetries and are collectively called accidental symmetries, all of which are continuous U(1) global symmetries. The transformations leaving the Lagrangian invariant are

U(1)

 

For the quantum # we require an operator that delivers the value for the quantum #

(see e.g. Moore pg 95). The operator and a parameter appear in the exponent so for particles that have a 0 or no value for the Q# there will be no change.

The accidental Q#s are Baryon #, electron #, muon # and Tau number. For baryon number all quark states are changed by a common phase. For the lepton #s only the individual states are changed as shown below.

 

E_L\rightarrow e^{i\beta}E_L\text{ and }(e_R)^c\rightarrow  e^{i\beta}(e_R)^c

M_L\rightarrow  e^{i\beta}M_L\text{ and }(\mu_R)^c\rightarrow  e^{i\beta}(\mu_R)^c

T_L\rightarrow  e^{i\beta}T_L\text{ and }(\tau_R)^c\rightarrow e^{i\beta}(\tau_R)^c.

The first transformation rule is shorthand to mean that all quark fields for all generations must be rotated by an identical phase simultaneously. The fields ML, TL and R)c, R)c are the 2nd (muon) and 3rd (tau) generation analogs of EL and (eR)c fields.

By Noether's theorem, each of these symmetries yields an associated conservation law. They are the conservation of baryon number, electron number, muon number, and tau number. Each quark carries 1/3 of a baryon number, while each antiquark carries -1/3 of a baryon number. The conservation law implies that the total number of quarks minus number of antiquarks stays constant throughout time. Within experimental limits, no violation of this conservation law has been found.

For discrete symmmetries there is a Q# for continuous symmetries there is a current that describes the movement of the quantum number and the integral of the 4th component is the conserved quantity. Consider electric currents charge move to change local densites and the integral of the charge denstity remains constant.

Similarly, each electron and its associated neutrino carries +1 electron number, while the antielectron and the associated antineutrino carry -1 electron number, the muons carry +1 muon number and the tau leptons carry +1 tau number. The standard model predicts that each of these three numbers should be conserved separately in a manner similar to the baryon number. These numbers are collectively known as lepton family numbers (LF). The difference in the symmetry structures between the quark and the lepton sectors is due to the masslessness of neutrinos in the standard model. However, it was recently found that neutrinos have small mass, and oscillate between flavors, signaling the violation of these three quantum numbers.

In addition to the accidental (but exact) symmetries described above, the standard model exhibits a set of approximate symmetries. These are the SU(2) Custodial Symmetry and the SU(2) or SU(3) quark flavor symmetry.

Left-handed fermions in the Standard Model

Generation 1

Fermion
(left-handed)

Symbol

Electric
charge

Weak
isospin

Weak
hypercharge

Color
charge
 *

Mass **

 

Electron

e^-\,

-1\,

-1/2\,

-1\,

\bold{1}\,

511 keV

 

Positron

e^+\,

+1\,

0\,

+2\,

\bold{1}\,

511 keV

 

Electron-neutrino

\nu_e\,

0\,

+1/2\,

-1\,

\bold{1}\,

< 2 eV ****

 

Up quark

u\,

+2/3\,

+1/2\,

+1/3\,

\bold{3}\,

~ 3 MeV ***

 

Up antiquark

\bar{u}\,

-2/3\,

0\,

-4/3\,

\bold{\bar{3}}\,

~ 3 MeV ***

 

Down quark

d\,

-1/3\,

-1/2\,

+1/3\,

\bold{3}\,

~ 6 MeV ***

 

Down antiquark

\bar{d}\,

+1/3\,

0\,

+2/3\,

\bold{\bar{3}}\,

~ 6 MeV ***

 

 

Generation 2

Fermion
(left-handed)

Symbol

Electric
charge

Weak
isospin

Weak
hypercharge

Color
charge *

Mass **

 

Muon

\mu^-\,

-1\,

-1/2\,

-1\,

\bold{1}\,

106 MeV

 

Antimuon

\mu^+\,

+1\,

0\,

+2\,

\bold{1}\,

106 MeV

 

Muon-neutrino

\nu_\mu\,

0\,

+1/2\,

-1\,

\bold{1}\,

< 2 eV ****

 

Charm quark

c\,

+2/3\,

+1/2\,

+1/3\,

\bold{3}\,

~ 1.337 GeV

 

Charm antiquark

\bar{c}\,

-2/3\,

0\,

-4/3\,

\bold{\bar{3}}\,

~ 1.3 GeV

 

Strange quark

s\,

-1/3\,

-1/2\,

+1/3\,

\bold{3}\,

~ 100 MeV

 

Strange antiquark

\bar{s}\,

+1/3\,

0\,

+2/3\,

\bold{\bar{3}}\,

~ 100 MeV

 

 

Generation 3

Fermion
(left-handed)

Symbol

Electric
charge

Weak
isospin

Weak
hypercharge

Color
charge *

Mass **

 

Tau lepton

\tau^-\,

-1\,

-1/2\,

-1\,

\bold{1}\,

1.78 GeV

 

Anti-tau lepton

\tau^+\,

+1\,

0\,

+2\,

\bold{1}\,

1.78 GeV

 

Tau-neutrino

\nu_\tau\,

0\,

+1/2\,

-1\,

\bold{1}\,

< 2 eV ****

 

Top quark

t\,

+2/3\,

+1/2\,

+1/3\,

\bold{3}\,

171 GeV

 

Top antiquark

\bar{t}\,

-2/3\,

0\,

-4/3\,

\bold{\bar{3}}\,

171 GeV

 

Bottom quark

b\,

-1/3\,

-1/2\,

+1/3\,

\bold{3}\,

~ 4.2 GeV

 

Bottom antiquark

\bar{b}\,

+1/3\,

0\,

+2/3\,

\bold{\bar{3}}\,

~ 4.2 GeV

 

Notes:

                 * These are not ordinary abelian charges, which can be added together, but are labels of group representations of Lie groups.

                 ** Mass is really a coupling between a left-handed fermion and a right-handed fermion. For example, the mass of an electron is really a coupling between a left-handed electron and a right-handed electron, which is the antiparticle of a left-handed positron. Also neutrinos show large mixings in their mass coupling, so it's not accurate to talk about neutrino masses in the flavor basis or to suggest a left-handed electron antineutrino.

                 *** The masses of baryons and hadrons and various cross-sections are the experimentally measured quantities. Since quarks can't be isolated because of QCD confinement, the quantity here is supposed to be the mass of the quark at the renormalization scale of the QCD scale.

                 **** The Standard Model assumes that neutrinos are massless. However, several contemporary experiments prove that neutrinos oscillate between their flavour states, which could not happen if all were massless. [11] It is straightforward to extend the model to fit these data but there are many possibilities, so the mass eigenstates are still open. See Neutrino#Mass.

Parity

Momentum switches

Spin unchanged

 

Examine a wheel spinning and spinning in a mirror. The object would spin in the same direction.

Consider the rotating wheel shown above. Map all the points on the wheel on the right to point on the wheel on the left through parity. These points are found by drawing a line through the origin (x,y,z -x,-y,-z). Now map these same points again at short time later. The result is a parity transformation acting on the object (not the coordinate system) preserves the sense of the rotation.

 

Angular momentum is an axial vector. No change in sign.

Position, velocity and momentum are vectors. Change sign.

 

Proper Poincare transformations are all the translations, rotations and boosts that are continuously connected to the identity. There is no way to define a transformation that is dependent on a parameter, such that changes in the parameter start the corresponding transformation at the identity and at some other parameter value the transformation is a parity transformation. 2-d rotations for θ=0 o correspond to the identity. The angle can be increased from 0 o to for example to 30 o through arbitrarily small steps. The rotation about an axis by 30o is therefore continuously connected back to the identity.

 

There are three important types of transformations that fall into this category:

  • Parity (x,y,z -x,-y,-z)
  • Charge Conjugation (particle antiparticle)
  • Time reversal (t -t)

Parity and time reversal are improper transformations but are part of the full Poincare set of possible transformations. Charge conjugation is an additional transformation that does not involve space-time but is essential for field theories.

 

How do states (experiments, systems) transform under parity?

It is convenient to find states that have special properties (eigenstates).

Even without QM, function space can be examined in terms of odd and even functions. In general a function is neither odd or even but and function can be written as a sum of odd and even.

 

[SAME EQUATION AS ABOVE]

 

F is a general function that can always be written in terms of some odd and some even function.

 

Since a particle will be described by its spatial wavefunction it is convenient to recognize that a general function can also be written in terms of R(r)Ylm(θ,φ), which are the angular momentum states.

To finalize our understanding of parity we need to ask if any of the internal (non-space-time) properties are impacted by a parity transformation.

 

 

Dirac equation (the relativistic treatment of the fermions) predicts, find, or demands that there be antiparticles related to particles. The relationship specifies that the parity of fermion and antifermion are opposite.

 

The solution for free fields based on Maxwells equations has an ambiguity that is usually resolved by choosing the Electric field to be a vector and the Magnetic field to be an axial vector. The four vector Aμ potential then has an overall negative parity (transforms as a vector).

 

Combining quarks to make composite system such as a pion requires the combination of the spatial, spin, intrinsic parity to find the overall parity of the composite system.