Symmetries and conservation Laws: Part 2

Baryon and Lepton numbers

• The Baryon number B=1 is assigned to all baryons and B=-1 is assigned to all anti baryons: all other particles have B=0.
• The lepton number L_e=1 is assigned to electrons and electron neutrino, L_e=-1 to their anti particles ; all other particles have L_e=0
• L_μ=1 for muon and μ-neutrino and L_τ=1 for tau lepton and its neutrino.
• Significance of these numbers is that , in every process of whwtever kind , the total valuse of B, L_e,L_μ,L_τ separately remains constant.
• Conservation of leptons has a signefence for strond interactions.
• Another property that is conserved only in strong interactions is isospin.

Strangeness

• A number of particles were discovered that behsve so unexpectedly that they were called strange particles.
• They were created in pairs, and decay only in certain ways but not in others that were allowed by existing conservation laws.
• To clarify the observation Gell-Mann and Nishijina indepensently introduced the strangeness number.
• For photon , π⁰ and η⁰, B, L_e,L_μ, L_τ and S are zero . There is no way to distinguish between them and their antiparticles, and they are regarded as their own anti particles.
• Strangeness number is conserved in all processes mediated by strong and electromagnetic interactions.
• The multiple creation of particles with S≠0 is the result of this conservation principle
• S can change in an event mediated by the week interaction. Decays that proceed via a week interaction are relatively slow, a billion tomes slower than the interactions proceeded via strong interactions.
• Week interactions does not allow S to change by more than ±1 in a decay. For example,

  Ξ^- decays in two steps Ξ^- →Λ⁰+π^-→η⁰+π⁰

Isospin

• There are number of hadron families whose numbers have similar masses but different charges. Thes families are called multiplets. Member of multiplet represents different charged states of a single fundemental entity.
• Each multiplet according to number of charge states exhibits a number I such that the multiplicity of state is given by 2I+1.
• Isospin can be represented by vector I in an abstract iso space whose component in any specific direction is governed by the quantum number denoted by I₃.
• Possible values of I₃ varies from I, I-1 to -I. The charge of a baryon is related to its baryon numberB, its strangeness number S and the component I₃ of its isotopic spin by the formula

  $Q=e(I_3+\frac{B}{2}+\frac{S}{2})$

Conservation of statistics

• The interchange of identicle particles in a system is a type of symmetry operation which leads to the preservation of the wave
• Conservation of statistics signifies that no process occuring within an isolated system can change the statistical behaviour of the system.

Hypercharge

• Hypercharge is defined as Y=S+B
• Classification system for hadrons encompasses many short lived particles as well as relatively stable hadrons
• This scheme cillects isospin multiplets into submultiplets whose members have the same spin but different in isospin and a quantity called hypercharge.

Quantum Mechanics (Uncertainity principle)

Question If a freely moving electron is localized in space to within $\Delta x_0$ of $x_0$, its wave function can be described by a wave packet $\psi(x,t)=\int_\infty^{-\infty}e^{i(kx-\omega t)}f(k)dk$, where $f(k)$ is peaked around a central value $k_0$. Which of the following is most nearly the width of the peak in $k$?
A. $\Delta k = 1/x_0$
B. $\Delta k = \frac{1}{\Delta x_0}$
C. $\Delta k = \frac{\Delta x_0}{x_0^2}$
D. $\Delta k = k_0\frac{\Delta x_0}{x_0}$
E. $\Delta k = \sqrt{k_0^2+(1/x_0)^2}$
Solution:
In quantum mechanics, the momentum $(p=\hbar{k})$ and position $(x)$ wave functions are Fourier transform pairs and the relation between $p$ and $x$ representations forms the Heisenberg uncertainty relation:
$\Delta{x}\Delta{k}\geq1 \Rightarrow \Delta k \geq \frac{1}{\Delta x}$
Answer: B

Symmetries and conservation Laws: Part 1

• Every conservation principle corresponds to symmetry in nature
• A symmetry of a particular kind exists when a certain operation leaves something unchanged.
• There is an intimate connection between symmetry and so called conserved quantities.
• Well known conserved quantity is energy and corresponding symmetry in this case is time translation.

Momentum Conservation

• Holds for all type of interactions
• Related to the invariance of physical laws under translation in space.
• Thus laws of interaction do not depend on the place of measurement so the space is homogeneous.
• This transnational uniformity of space leads to the conservation of linear momentum.
• Particle at rest have no momentum. If it  decays into two less massive particles , momentum conservation requires that the two particles travel away in exactly opposite directions.

Conservation of Energy

• Holds for all type of interactions.
• related to the invariance of physical laws under translations along the time axis i.e., homogeneity of time.
• laws of interaction do not depend on the time of measurement

Angular momentum conservation

• In addition to transnational symmetry , space also has a rotational symmetry.
• This symmetry of space gives rise to another conserved quantity , angular momentum.
• This law is also of general validity for all types of interactions.
• It is related to the invariance of the physical laws under rotation (isotropy of space).
• The orbital and spin angular momentum may be separately conserved.

Parity Conservation

• Holds for strong, nuclear and electromagnetic interactions but is violated in case of week interactions.
• related to the invariance of the physical laws under inversion of space co-ordinates. x,y,z are replaced by -x.-y,-z.
• is equivalent to combined reflection and rotation.
• physical laws do not depend on the right handedness of co-ordinate system.
• Parity operation symmetry represents discrete symmetry (reflection and rotation through 180 degree)
• Every particle with non zero mass has an intrinsic parity  π which can either be +1(even) or -1 (odd). Thus total parity of a system of n particles is the product of their intrinsic parities and the orbital parity (-1)^l.
Thus, π_tot=π₁π₂π₃.......π_n(-1)^l
• Intrinsic parity of pions is odd.

Conservation of charge

• Conservation of electric charge is related to gauge transformations which are shifts in the zeros of the scalar and vector electromagnetic potentials V and A
• Gauge transformations leave E and B unaffected since the latter are obtained by differentiating potentials , and this invariance leads to charge conservation.
• Charge and baryon number are conserved in all interactions.