Particle physics Flashcards
What are fundamental particles?
Point-like objects without any internal structure
What are intrinsic properties? (not examples)
Properties that are independent of reference frame
What are examples of intrinsic properties of particles?
Mass, spin, charge (type changes depending on the type of interaction), parity and charge conjugation
Why do we need to be careful about the mass for unstable particles?
The mass (and energy) aren’t fixed if the uncertainty in time is not infinity because of Heisenberg’s uncertainty principle
How is the width (capital gamma) related to the mean lifetime of the particle (tau)?
The mean lifetime is equal to h bar over the width
What are particles with half-integer spin?
Fermions
What are particle with integer spin?
Bosons
What is general charge?
It is the quantum number that tells us how likely a particle interacts via a particular force (ie EM, weak, strong, gravitational)
What are the extrinsic properties of a particle? (ones that do depend on the reference frame)
Four momentum and the projection of the spin onto some axis (like helicity)
What is the helicity?
Two times the projection of the spin onto the axis defined by the particle’s momentum
What is the amplitude when referring to Feynman diagrams?
A path a particle can take and the modulus squared of it gives the probability
What is the matrix element when considering Feynman diagrams and what is also known as?
It is the total complex amplitude for a transition from one initial state to a final state and also known as the transition amplitude
What is the transition rate?
It is proportional to the magnitude squared of the matrix element (for it to be equal to rather than proportional, there is a phase space term as well)
What we will be ignoring in Feynman diagrams?
Spin
What is conserved at each vertex?
Energy, momentum (including 4 momentum) and charge
What direction does time run on a Feynman diagram?
Left to right (LHS is initial state, RHS is final state and middle is how it happened)
What direction do anti-particles point on Feynman diagrams?
In the negative time direction
What does each vertex of a Feynman diagram contribute and what is it proportional to?
A coupling strength factor g and the charge of the particle (charge changes depending on the type of interaction, eg electric charge for EM etc)
In a Feynman diagram, what factor does each intermediate particle (propagator) contribute?
1 over q squared minus m squared (q = four momentum)
Is the intermediate particle in Feynman diagrams real or virtual and is it on or off shell?
Virtual and off shell
What is the contribution from a massless propagator (like a photon)?
1 over E squared minus p squared
What are 3 types of Lorentz transformations?
Rotations, shift and boosts
What do Lorentz transformations do?
They transform between two frames that have a constant velocity relative to each other
For a Lorentz transformation that is a rotation about one of the axes, what parts of the 4 component are not affected?
Time and the axis it is being rotated about
For a Lorentz transformation that is a boost in either the x, y or z direction, what does the matrix look like?
Along the diagonal it always has gamma first then the rest of the diagonal are 1 except the one that will correspond to the particular direction will be a gamma. The rest are zeros except the first one in the row and column that correspond with x (2nd row and column), y (3rd) and z (4th)
What is a shift/translation in Lorentz transformation?
dx is added to x (general)
What are four-vectors?
Objects that transform under Lorentz transformations like the differences in ct, x, y, z
What is the energy using Lorentz transformations?
gamma multiplied by mass times c squared
What is the modulus of the momentum vector using Lorentz transformations?
beta times gamma times mass times c
What is beta?
The speed of the particles
What is gamma?
1 over square root 1 minus beta squared
What is the dot products between two four vectors?
the 0th components (first part of the vector) of the vectors multiplied together minus the dot product of the 3 vector (last 3 components of the vectors)
What does Lorentz invariant mean?
Independent of reference frame/ the same in any coordinate system
Are dot products Lorentz invariant?
Yes
What is the centre of mass energy?
root s
What is tau, the proper time of the particle?
The same as the time of the particle at rest
What is time dilation?
Time intervals observed from a moving coordinate system are stretched by a factor of gamma relative to the time in the rest frame
What are the terms in the p four vector?
E is the first term, and the other three are the speed of light (c) multiplied by the momentum in that direction eg cp(x), cp(y), cp(z)
What properties are conserved quantities at any point in space and time?
Energy and momentum
What quantity is not conserved but is invariant under Lorentz transformations?
Mass
What do we change when using natural units?
The speed of light = h bar =1
What is the main problems of Schrodinger’s equation?
It does not account for relativistic effects and isn’t Lorentz invariant (looks different in different inertial frames)
What does the sum of the differential of the probability density with respect to time and the divergence of the probability current equal?
Zero
What is the Klein-Gordan equation?
It is a relativistic wave equation, related to Schrodinger’s equation
What was the problem with the Klein-Gordan equation?
The solutions to the equation can give negative probability density (and negative energy solution)
What are the gamma matric elements in the Dirac equation?
They are four 4x4 matrices that look like 2x2 matrices but each component is itself another 2x2 matrix
Is the probability density always positive positive for the Dirac equation?
Yes
The negative energy solutions to the Dirac equation are explained by Feynman and Stueckelberg as being what?
Positive energy energy anti-particles travelling backwards in time
How many solutions are there to the Dirac equation and what type of particle do they respond to?
4 and the first two correspond to particle solutions and the final 2 correspond to anti-particle solutions
The solutions to the Dirac equation are also eigenstates of what?
The Hamiltonian operator
What are the components of total angular momentum (J) which means it is conserved?
Orbital angular momentum (L) and the spin operator (sigma)
What does the spin operators look like to be suitable with the four components of the Dirac equation solutions?
4D matrices with the 2x2 Pauli matrices included in 2x2 matrices
How are the Hamiltonian, three momentum, angular momentum and spin operators for antiparticles related to that of particles?
They are the negatives of them
The Dirac equation describes particles with how much spin in the z direction?
Plus or minus half
Is the helicity quantum number conserved and what commutation relation shows this?
The Hamiltonian commutes with helicity
Is helicity Lorentz invariant?
No
What are the eigenvalues of the helicity operator for a spin-half particle?
Plus or minus one
What is the total spin operator equal to?
The spin operator multiplied by its adjoint
When is the only cases when the wavefunctions that are solutions to the Dirac equation are eigenstates of the z direction spin operator?
The particle is at rest or only carries momentum along the z- axis (not the case in general though)
Are the wavefunctions that are solutions to the Dirac equation eigenstates of the total spin operator?
Yes
What are the eigenvalues of the total spin operator with the wavefunctions that are solutions to the Dirac equation as eigenstates?
3/4 (three quarters)
Why must the force (the strong force) holding the nucleons together be short range?
Nucleons can be observed as free particles, without needing to bond into a nucleus
What did Yukawa suggest?
The force holding nucleons together must be short range and the carrier of the strong force must be massive. He predicted the mass of this particle to be 200x that of an electron and a tenth of a proton and called it a meson
What is the Yukawa meson?
A pion
What is a Yukawa interaction?
The nuclear force between nucleons mediated by pions in low energy regimes
The fact that the mass of protons is approx the same as that of neutrons and that the elastic scattering of protons and neutrons have the same coupling strengths means what?
The strong force acts the same on protons and neutrons
What is strong isospin and what are its components?
It is the quantum number related to the up and down quark content of the particle. The first is its isospin and the second is its 3rd-component isospin (I subscript 3)
What is the value of the isospin?
1/2 for protons and neutrons and 1 for pions
What is the value for the third component of isospin?
Plus 1/2 for up quarks and minus 1/2 for down quarks and 0 for other quarks
What is an isospin-doublet and what is an example of one?
Two different states with different isospins of a single isospin, like a neutron and a proton make up a nucleon
What are the isospin values of protons and neutrons?
Protons: |1/2,1/2> and neutrons: |1/2, -1/2>
What are the isospin values of the isospin-triplet made by the three types of pions?
The first value (isospin) is 1 and the third component isospin is the same as its charge (eg +1 for the positive pion, 0 for neutral etc)
If the strong force is symmetric under isospin transformation, what does this mean by Noether’s theorem?
Isospin is conserved in isospin interactions
How do we calculate the total isospin with 2 particles?
The total isospin is between the sum of the particles isospins and the modulus of the difference between the particles isospins. Also, it is constrained by the fact that the total third component has to be between the total isospin and minus the total isospin
How do we calculate the total 3rd component isospin with 2 particles?
Just the addition of both third components of each particle.
Does the strong force treat up and down quarks equally?
Yes
What is the strong isospin of up and down quarks?
Both have 1/2 as the first component with 1/2 for the third component of the up quark and -1/2 for the third component of the down quark