Attempts at relativistic QM
|
|
- Kristopher Bradley
- 5 years ago
- Views:
Transcription
1 Attempts at relativistic QM based on S-1 A proper description of particle physics should incorporate both quantum mechanics and special relativity. However historically combining quantum mechanics and relativity was highly non-trivial. Today we review some of these attempts. The result of this effort is relativistic quantum field theory - consistent description of particle physics.
2 Quantum mechanics Time evolution of the state of the system is described by Schrödinger equation: where H is the hamiltonian operator representing the total energy. For a free, spinless, nonrelativistic particle we have: in the position basis, P = ih H = 1 2m P2 and S.E. is: where ψ(x, t) = x ψ, t is the position-space wave function.
3 Relativistic generalization? Obvious guess is to use relativistic energy-momentum relation: Schrödinger equation becomes: Not symmetric in time and space derivatives
4 Relativistic generalization? Obvious guess is to use relativistic energy-momentum relation: Schrödinger equation becomes: Applying i t on both sides and using S.E. we get Klein-Gordon equation looks symmetric
5 Special relativity (physics is the same in all inertial frames) Space-time coordinate system: x µ = (ct, x) define: x µ = ( ct, x) or where is the Minkowski metric tensor. Its inverse is: that allows us to write:
6 Interval between two points in space-time can be written as: ds 2 = (x x ) 2 c 2 (t t ) 2 = g µν (x x ) µ (x x ) ν = (x x ) µ (x x ) µ General rules for indices: repeated indices, one superscript and one subscript are summed; these indices are said to be contracted any unrepeated indices (not summed) must match in both name and height on left and right side of any valid equation
7 Two coordinate systems (representing inertial frames) are related by Lorentz transformation matrix translation vector Interval between two different space-time points is the same in all inertial frames: which requires:
8 Notation for space-time derivatives: matching-index-height rule works: For two coordinate systems related by derivatives transform as: which follows from not the same as will prove later!
9 Is K-G equation consistent with relativity? Physics is the same in all inertial frames: the value of the wave function at a particular space-time point measured in two inertial frames is the same: This should be true for any point in the space-time and thus a consistent equation of motion should have the same form in any inertial frame. Is that the case for Klein-Gordon equation?
10 Klein-Gordon equation: in 4-vector notation: in 4-vector notation: = d Alambertian operator Is it equivalent to: Since? K-G eq. is manifestly consistent with relativity!
11 Is K-G consistent with quantum mechanics? Schrödinger equation (first order in time derivative) leaves the norm of a state time independent. Probability is conserved: = d 3 xρ(x) ψ, t ψ, t = t d 3 x ρ t = d 3 x. j = S j.ds = 0 ρ t = ψ ψ t + ψ ψ t = i 2m ψ 2 ψ i 2m ψ 2 ψ = i 2m.(ψ ψ ψ ψ ). j Gauss s law j(x) = 0 at infinity
12 Is K-G consistent with quantum mechanics? Schrödinger equation (first order in time derivative) leaves the norm of a state time independent. Probability is conserved: = d 3 xρ(x) ψ, t ψ, t = t d 3 x ρ t = d 3 x. j = S j.ds = 0 Klein-Gordon equation is second order in time derivative and the norm of a state is NOT in general time independent. Probability is not conserved. Klein-Gordon equation is consistent with relativity but not with quantum mechanics.
13 Dirac attempt Dirac suggested the following equation for spin-one-half particles (a state carries a spin label, a = 1,2): ψ, a, t Consistent with ψ(x, Schrödinger t) = x ψ, equation t for the Hamiltonian: Dirac equation is linear in both time and space derivatives and so it might be consistent with both QM and relativity. Squaring the Hamiltonian yields:
14 can be written as anticommutator and also can be written as H 2 Eigenvalues of ψ(x, t) should = x ψ, satisfy t the correct relativistic energymomentum relation: and so we choose matrices that satisfy following conditions: it can be proved (later) that the Dirac equation is fully consistent with relativity. we have a relativistic quantum mechanical theory!
15 Discussion of Dirac equation to account for the spin of electron, the matrices should be 2x2 but the minimum size satisfying above conditions is 4x4 - two extra spin states H is traceless, and so 4 eigenvalues are: E(p), E(p), -E(p), -E(p) negative energy states no ground state also a problem for K.-G. equation Dirac s interpretation: due to Pauli exclusion principle each quantum state can be occupied by one electron and we simply live in a universe with all negative energy states already occupied. Negative energy electrons can be excited into a positive energy state (by a photon) leaving behind a hole in the sea of negative energy electrons. The hole has a positive charge and positive energy antiparticle (the same mass, opposite charge) called positron (1927)
16 Quantum mechanics as a quantum field theory Consider Schrödinger equation for n particles with mass m, moving in an external potential U(x), with interparticle potential It is equivalent to: is the position-space wave function. if and only if satisfies S.E. for wave function!
17 and is a quantum field and its hermitian conjugate; they satisfy commutation relations: is the vacuum state state with one particle at state with one particles at and another at Total number of particles is counted by the operator: commutes with H!
18 creation operators commute with each other, and so without loss of generality we can consider only completely symmetric functions: we have a theory of bosons (obey Bose-Einstein statistics). For a theory of fermions (obey Fermi-Dirac statistics) we impose and we can restrict our attention to completely antisymmetric functions: We have nonrelativistic quantum field theory for spin zero particles that can be either bosons or fermions. This will change for relativistic QFT.
19 Lorentz invariance based on S-2 Lorentz transformation (linear, homogeneous change of coordinates): that preserves the interval : All Lorentz transformations form a group: product of 2 LT is another LT identity transformation: inverse: for inverse can be used to prove:
20 Infinitesimal Lorentz transformation: thus there are 6 independent ILTs: 3 rotations and 3 boosts not all LT can be obtained by compounding ILTs! +1 proper -1 improper proper LTs form a subgroup of Lorentz group; ILTs are proper! Another subgroup - orthochronous LTs, ILTs are orthochronous!
21 When we say theory is Lorentz invariant we mean it is invariant under proper orthochronous subgroup only (those that can be obtained by compounding ILTs) Transformations that take us out of proper orthochronous subgroup are parity and time reversal: orthochronous but improper nonorthochronous and improper A quantum field theory doesn t have to be invariant under P or T.
22 How do operators and quantum fields transform? Lorentz transformation (proper, orthochronous) is represented by a unitary operator that must obey the composition rule: infinitesimal transformation can be written as: are hermitian operators = generators of the Lorentz group from using and expanding both sides, keeping only linear terms in we get: since are arbitrary general rule: each vector index undergoes its own Lorentz transformation!
23 using and expanding to linear order in we get: These comm. relations specify the Lie algebra of the Lorentz group. We can identify components of the angular momentum and boost operators: and find:
24 in a similar way for the energy-momentum four vector we find: P µ = (H/c, P i ) using and expanding to linear order in we get: or in components: in addition: Comm. relations for J, K, P, H form the Lie algebra of the Poincare group.
25 Finally, let s look at transformation of a quantum scalar field: Recall time evolution in Heisenberg picture: this is generalized to: P x = P µ x µ = P x Ht x is just a label we can write the same formula for x-a: e +ip a/ e ip x/ φ(0)e +ip x/ e ip a/ = φ(x a) we define space-time translation operator: and obtain:
26 Similarly: Derivatives carry vector indices: is Lorentz invariant
27 Canonical quantization of scalar fields Hamiltonian for free nonrelativistic particles: based on S-3 Fourier transform: we get: a(x) = d 3 p (2π) 3/2 eip x ã(p) d 3 x (2π) 3 eip x = δ 3 (p) can go back to x using: d 3 p (2π) 3 eip x = δ 3 (x)
28 Canonical quantization of scalar fields (Anti)commutation relations: Vacuum is annihilated by : [A, B] = AB BA is a state of momentum, eigenstate of with is eigenstate of with energy eigenvalue:
29 Relativistic generalization Hamiltonian for free relativistic particles: Is this theory Lorentz invariant? spin zero, but can be either bosons or fermions Let s prove it from a different direction, direction that we will use for any quantum field theory from now: start from a Lorentz invariant lagrangian or action derive equation of motion (for scalar fields it is K.-G. equation) find solutions of equation of motion show the Hamiltonian is the same as the one above
30 A theory is described by an action: where is the lagrangian. Equations of motion should be local, and so where is the lagrangian density. Thus: is Lorentz invariant: For the action to be invariant we need: the lagrangian density must be a Lorentz scalar!
31 Any polynomial of a scalar field is a Lorentz scalar and so are products of derivatives with all indices contracted. Let s consider: arbitrary constant = 1, c = 1 and let s find the equation of motion, Euler-Lagrange equation: (we find eq. of motion from variation of an action: making an infinitesimal variation in and requiring the variation of the action to vanish) integration by parts, and δφ(x) = 0 at infinity in any direction (including time) is arbitrary function of x and so the equation of motion is Klein-Gordon equation
32 Solutions of the Klein-Gordon equation: one classical solution is a plane wave: is arbitrary real wave vector and The general classical solution of K-G equation: where and are arbitrary functions of, and is a function of k (introduced for later convenience) if we tried to interpret as a quantum wave function, the second term would represent contributions with negative energy to the wave function!
33 real solutions: k k thus we get: (such a k µ is said to be on the mass shell)
34 Finally let s choose so that is Lorentz invariant: manifestly invariant under orthochronous Lorentz transformations on the other hand sum over zeros of g, in our case the only zero is k 0 = ω for any the differential is Lorentz invariant it is convenient to take for which the Lorentz invariant differential is:
35 Finally we have a real classical solution of the K.-G. equation: where again:,, For later use we can express in terms of : where and we will call. Note, is time independent.
36 Constructing the hamiltonian: Recall, in classical mechanics, starting with lagrangian as a function of coordinates and their time derivatives we define conjugate momenta and the hamiltonian is then given as: In field theory: and the hamiltonian is given as: hamiltonian density
37 In our case: Inserting we get:
38 d 3 x (2π) 3 eip x = δ 3 (p)
39 From classical to quantum (canonical quantization): coordinates and momenta are promoted to operators satisfying canonical commutation relations: operators are taken at equal times in the Heisenberg picture
40 We have derived the classical hamiltonian: We kept ordering of a s unchanged, so that we can easily generalize it to quantum theory where classical functions will become operators that may not commute. The hamiltonian of the quantum theory: (2π) 3 δ 3 (0) = V see the formula for delta function is the total zero point energy per unit volume we are free to choose: the ground state has zero energy eigenvalue.
41 Summary: is equivalent to: for: We have rederived the hamiltonian of free relativistic bosons by quantization of a scalar field whose equation of motion is the Klein- Gordon equation (starting with manifestly Lorentz invariant lagrangian). does not work for fermions, anticommutators lead to trivial hamiltonian!
42 The spin-statistics theorem Hamiltonian of free spin zero particles: based on S-4 How to add Lorentz invariant interactions?
43 Let s split the hermitian free field into: where: time evolved with : For Lorentz transformation (proper, orthochronous) we have found: Thus and are Lorentz scalars. We will have local, Lorentz invariant interactions if we take the interaction lagrangian density to be a hermitian function of and.
44 Time dependent perturbation theory in QM: the transition amplitude to start with an initial state at time and end with a final state at time is: where is the perturbing hamiltonian in the interaction picture, where is the perturbing hamiltonian in the Schrödinger picture. T is the time ordering symbol a product of operators to its right is to be ordered, not as written, but with operators at later times to the left of those at earlier times. Specifying : H 1 (x, 0) = F(ϕ + (x, 0), ϕ (x, 0)) H I (x, t) = F(ϕ + (x, t), ϕ (x, t))
45 Time evolution operator: satisfies: α, t = U I (t, t 0 ) α, t 0 i t U I(t, t 0 ) = H I (t)u I (t, t 0 ) with initial condition: U I (t 0, t 0 ) = 1 It is equivalent to the integral equation: U I (t, t 0 ) = 1 i t t 0 H I (t )U I (t, t 0 )dt which we can solve iteratively: U I (t, t 0 ) = 1 i t t 0 H I (t ) 1 i t t 0 H I (t )U I (t, t 0 )dt dt
46 U I (t, t 0 ) = 1 i H I (t ) 1 i t 0 t t t 0 H I (t )U I (t, t 0 )dt dt more iterations: using: U I (t, t 0 ) = 1 + ( i) t t +( i) 2 dt t 0 t 0 dt H I (t ) + ( i) 2 t t 0 t dt t t t t t t dt dt H I (t )H I (t ) t 0 t 0 t 0 dt H I (t )H I (t )H I (t ) + dt dt H I (t )H I (t ) = 1 dt dt T {H I (t )H I (t )} t 0 t 0 2 t 0 t 0 and with n! for higher order terms, we get: U I (t, t 0 ) = 1 + ( i) T exp t dt H I (t ) + ( i)2 t 0 2! t i dt H I (t ) t 0 t t dt dt T {H I (t )H I (t )} + t 0 t 0
47 In order for the transition amplitude ordering must be frame independent! to be Lorentz invariant, the time Time ordering of two space-time points is: frame independent if their separation is timelike: frame dependent if their separation is spacelike: Thus for we require: But not in general satisfied! modified Bessel function nonzero for any r > 0 even for m = 0!
48 Let s try using only particular linear combinations: we then have: and both can be 0 only if we choose commutators and Consistent describing interacting spin-zero particles can be obtained only with commutators. Spin-zero particles are bosons.
49 In addition: is hermitian! we can replace which doesn t modify commutation relations and leads to: Thus we have found that is the fundamental object we have to use to build a consistent lagrangian. And we have to use commutators. If we choose anticommutators, we find, which leads to a trivial lagrangian that is at most linear in fields.
Quantum Field Theory
Quantum Field Theory PHYS-P 621 Radovan Dermisek, Indiana University Notes based on: M. Srednicki, Quantum Field Theory 1 Attempts at relativistic QM based on S-1 A proper description of particle physics
More informationQuantum Field Theory
Quantum Field Theory PHYS-P 621 Radovan Dermisek, Indiana University Notes based on: M. Srednicki, Quantum Field Theory 1 Attempts at relativistic QM based on S-1 A proper description of particle physics
More informationQuantum Field Theory
Quantum Field Theory PHYS-P 621 Radovan Dermisek, Indiana University Notes based on: M. Srednicki, Quantum Field Theory 1 Attempts at relativistic QM based on S-1 A proper description of particle physics
More informationThe Klein-Gordon equation
Lecture 8 The Klein-Gordon equation WS2010/11: Introduction to Nuclear and Particle Physics The bosons in field theory Bosons with spin 0 scalar (or pseudo-scalar) meson fields canonical field quantization
More informationQFT. Unit 1: Relativistic Quantum Mechanics
QFT Unit 1: Relativistic Quantum Mechanics What s QFT? Relativity deals with things that are fast Quantum mechanics deals with things that are small QFT deals with things that are both small and fast What
More informationContinuous symmetries and conserved currents
Continuous symmetries and conserved currents based on S-22 Consider a set of scalar fields, and a lagrangian density let s make an infinitesimal change: variation of the action: setting we would get equations
More informationwhere P a is a projector to the eigenspace of A corresponding to a. 4. Time evolution of states is governed by the Schrödinger equation
1 Content of the course Quantum Field Theory by M. Srednicki, Part 1. Combining QM and relativity We are going to keep all axioms of QM: 1. states are vectors (or rather rays) in Hilbert space.. observables
More informationQuantum Physics 2006/07
Quantum Physics 6/7 Lecture 7: More on the Dirac Equation In the last lecture we showed that the Dirac equation for a free particle i h t ψr, t = i hc α + β mc ψr, t has plane wave solutions ψr, t = exp
More information1 Quantum fields in Minkowski spacetime
1 Quantum fields in Minkowski spacetime The theory of quantum fields in curved spacetime is a generalization of the well-established theory of quantum fields in Minkowski spacetime. To a great extent,
More informationParticle Physics Dr. Alexander Mitov Handout 2 : The Dirac Equation
Dr. A. Mitov Particle Physics 45 Particle Physics Dr. Alexander Mitov µ + e - e + µ - µ + e - e + µ - µ + e - e + µ - µ + e - e + µ - Handout 2 : The Dirac Equation Dr. A. Mitov Particle Physics 46 Non-Relativistic
More informationLecture 10. The Dirac equation. WS2010/11: Introduction to Nuclear and Particle Physics
Lecture 10 The Dirac equation WS2010/11: Introduction to Nuclear and Particle Physics The Dirac equation The Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist
More informationParticle Physics. Michaelmas Term 2011 Prof. Mark Thomson. Handout 2 : The Dirac Equation. Non-Relativistic QM (Revision)
Particle Physics Michaelmas Term 2011 Prof. Mark Thomson + e - e + - + e - e + - + e - e + - + e - e + - Handout 2 : The Dirac Equation Prof. M.A. Thomson Michaelmas 2011 45 Non-Relativistic QM (Revision)
More informationPlan for the rest of the semester. ψ a
Plan for the rest of the semester ϕ ψ a ϕ(x) e iα(x) ϕ(x) 167 Representations of Lorentz Group based on S-33 We defined a unitary operator that implemented a Lorentz transformation on a scalar field: and
More informationQuantum Field Theory Notes. Ryan D. Reece
Quantum Field Theory Notes Ryan D. Reece November 27, 2007 Chapter 1 Preliminaries 1.1 Overview of Special Relativity 1.1.1 Lorentz Boosts Searches in the later part 19th century for the coordinate transformation
More informationNotes on Quantum Field Theory. Mark Srednicki UCSB
March 2003 Notes on Quantum Field Theory Mark Srednicki UCSB Notes for the first quarter of a QFT course, based mostly on ϕ 3 theory in six dimensions. Please send any comments or corrections to mark@physics.ucsb.edu
More informationExtending the 4 4 Darbyshire Operator Using n-dimensional Dirac Matrices
International Journal of Applied Mathematics and Theoretical Physics 2015; 1(3): 19-23 Published online February 19, 2016 (http://www.sciencepublishinggroup.com/j/ijamtp) doi: 10.11648/j.ijamtp.20150103.11
More informationRepresentations of Lorentz Group
Representations of Lorentz Group based on S-33 We defined a unitary operator that implemented a Lorentz transformation on a scalar field: How do we find the smallest (irreducible) representations of the
More informationQuantum Electrodynamics Test
MSc in Quantum Fields and Fundamental Forces Quantum Electrodynamics Test Monday, 11th January 2010 Please answer all three questions. All questions are worth 20 marks. Use a separate booklet for each
More informationΑΒΓΔΕΖΗΘΙΚΛΜΝΞΟΠΡΣΤΥΦΧΨΩ αβγδεζηθικλμνξοπρςστυφχψω +<=>± ħ
CHAPTER 1. SECOND QUANTIZATION In Chapter 1, F&W explain the basic theory: Review of Section 1: H = ij c i < i T j > c j + ij kl c i c j < ij V kl > c l c k for fermions / for bosons [ c i, c j ] = [ c
More informationCHAPTER 1. SPECIAL RELATIVITY AND QUANTUM MECHANICS
CHAPTER 1. SPECIAL RELATIVITY AND QUANTUM MECHANICS 1.1 PARTICLES AND FIELDS The two great structures of theoretical physics, the theory of special relativity and quantum mechanics, have been combined
More informationMaxwell s equations. electric field charge density. current density
Maxwell s equations based on S-54 Our next task is to find a quantum field theory description of spin-1 particles, e.g. photons. Classical electrodynamics is governed by Maxwell s equations: electric field
More informationRelativistic Waves and Quantum Fields
Relativistic Waves and Quantum Fields (SPA7018U & SPA7018P) Gabriele Travaglini December 10, 2014 1 Lorentz group Lectures 1 3. Galileo s principle of Relativity. Einstein s principle. Events. Invariant
More informationQuantum Field Theory II
Quantum Field Theory II T. Nguyen PHY 391 Independent Study Term Paper Prof. S.G. Rajeev University of Rochester April 2, 218 1 Introduction The purpose of this independent study is to familiarize ourselves
More informationMaxwell s equations. based on S-54. electric field charge density. current density
Maxwell s equations based on S-54 Our next task is to find a quantum field theory description of spin-1 particles, e.g. photons. Classical electrodynamics is governed by Maxwell s equations: electric field
More informationSpin Statistics Theorem
What is? University of Chicago, Department of Physics May 11, 2008 What is? Organization of this talk Contents of and a little history A few heuristic methods to prove the theorem Elementary proof Understand
More information2 Quantization of the scalar field
22 Quantum field theory 2 Quantization of the scalar field Commutator relations. The strategy to quantize a classical field theory is to interpret the fields Φ(x) and Π(x) = Φ(x) as operators which satisfy
More informationThe Dirac equation. L= i ψ(x)[ 1 2 2
The Dirac equation Infobox 0.1 Chapter Summary The Dirac theory of spinor fields ψ(x) has Lagrangian density L= i ψ(x)[ 1 2 1 +m]ψ(x) / 2 where/ γ µ µ. Applying the Euler-Lagrange equation yields the Dirac
More information2.4 Parity transformation
2.4 Parity transformation An extremely simple group is one that has only two elements: {e, P }. Obviously, P 1 = P, so P 2 = e, with e represented by the unit n n matrix in an n- dimensional representation.
More informationReview and Notation (Special relativity)
Review and Notation (Special relativity) December 30, 2016 7:35 PM Special Relativity: i) The principle of special relativity: The laws of physics must be the same in any inertial reference frame. In particular,
More informationThe Dirac Field. Physics , Quantum Field Theory. October Michael Dine Department of Physics University of California, Santa Cruz
Michael Dine Department of Physics University of California, Santa Cruz October 2013 Lorentz Transformation Properties of the Dirac Field First, rotations. In ordinary quantum mechanics, ψ σ i ψ (1) is
More informationLSZ reduction for spin-1/2 particles
LSZ reduction for spin-1/2 particles based on S-41 In order to describe scattering experiments we need to construct appropriate initial and final states and calculate scattering amplitude. Summary of free
More informationREVIEW REVIEW. A guess for a suitable initial state: Similarly, let s consider a final state: Summary of free theory:
LSZ reduction for spin-1/2 particles based on S-41 In order to describe scattering experiments we need to construct appropriate initial and final states and calculate scattering amplitude. Summary of free
More informationPart I. Many-Body Systems and Classical Field Theory
Part I. Many-Body Systems and Classical Field Theory 1. Classical and Quantum Mechanics of Particle Systems 3 1.1 Introduction. 3 1.2 Classical Mechanics of Mass Points 4 1.3 Quantum Mechanics: The Harmonic
More informationSummary of free theory: one particle state: vacuum state is annihilated by all a s: then, one particle state has normalization:
The LSZ reduction formula based on S-5 In order to describe scattering experiments we need to construct appropriate initial and final states and calculate scattering amplitude. Summary of free theory:
More informationmsqm 2011/8/14 21:35 page 189 #197
msqm 2011/8/14 21:35 page 189 #197 Bibliography Dirac, P. A. M., The Principles of Quantum Mechanics, 4th Edition, (Oxford University Press, London, 1958). Feynman, R. P. and A. P. Hibbs, Quantum Mechanics
More informationPart III. Interacting Field Theory. Quantum Electrodynamics (QED)
November-02-12 8:36 PM Part III Interacting Field Theory Quantum Electrodynamics (QED) M. Gericke Physics 7560, Relativistic QM 183 III.A Introduction December-08-12 9:10 PM At this point, we have the
More information3.3 Lagrangian and symmetries for a spin- 1 2 field
3.3 Lagrangian and symmetries for a spin- 1 2 field The Lagrangian for the free spin- 1 2 field is The corresponding Hamiltonian density is L = ψ(i/ µ m)ψ. (3.31) H = ψ( γ p + m)ψ. (3.32) The Lagrangian
More informationPath integral in quantum mechanics based on S-6 Consider nonrelativistic quantum mechanics of one particle in one dimension with the hamiltonian:
Path integral in quantum mechanics based on S-6 Consider nonrelativistic quantum mechanics of one particle in one dimension with the hamiltonian: let s look at one piece first: P and Q obey: Probability
More informationLecture 5. Hartree-Fock Theory. WS2010/11: Introduction to Nuclear and Particle Physics
Lecture 5 Hartree-Fock Theory WS2010/11: Introduction to Nuclear and Particle Physics Particle-number representation: General formalism The simplest starting point for a many-body state is a system of
More informationQuantization of scalar fields
Quantization of scalar fields March 8, 06 We have introduced several distinct types of fields, with actions that give their field equations. These include scalar fields, S α ϕ α ϕ m ϕ d 4 x and complex
More informationLorentz Invariance and Second Quantization
Lorentz Invariance and Second Quantization By treating electromagnetic modes in a cavity as a simple harmonic oscillator, with the oscillator level corresponding to the number of photons in the system
More informationLecture 4 - Dirac Spinors
Lecture 4 - Dirac Spinors Schrödinger & Klein-Gordon Equations Dirac Equation Gamma & Pauli spin matrices Solutions of Dirac Equation Fermion & Antifermion states Left and Right-handedness Non-Relativistic
More informationIntroduction to Modern Quantum Field Theory
Department of Mathematics University of Texas at Arlington Arlington, TX USA Febuary, 2016 Recall Einstein s famous equation, E 2 = (Mc 2 ) 2 + (c p) 2, where c is the speed of light, M is the classical
More informationIntroduction to string theory 2 - Quantization
Remigiusz Durka Institute of Theoretical Physics Wroclaw / 34 Table of content Introduction to Quantization Classical String Quantum String 2 / 34 Classical Theory In the classical mechanics one has dynamical
More information3 Quantization of the Dirac equation
3 Quantization of the Dirac equation 3.1 Identical particles As is well known, quantum mechanics implies that no measurement can be performed to distinguish particles in the same quantum state. Elementary
More information11 Spinor solutions and CPT
11 Spinor solutions and CPT 184 In the previous chapter, we cataloged the irreducible representations of the Lorentz group O(1, 3. We found that in addition to the obvious tensor representations, φ, A
More informationThe Dirac Equation. Topic 3 Spinors, Fermion Fields, Dirac Fields Lecture 13
The Dirac Equation Dirac s discovery of a relativistic wave equation for the electron was published in 1928 soon after the concept of intrisic spin angular momentum was proposed by Goudsmit and Uhlenbeck
More information7 Quantized Free Dirac Fields
7 Quantized Free Dirac Fields 7.1 The Dirac Equation and Quantum Field Theory The Dirac equation is a relativistic wave equation which describes the quantum dynamics of spinors. We will see in this section
More informationMSci EXAMINATION. Date: XX th May, Time: 14:30-17:00
MSci EXAMINATION PHY-415 (MSci 4242 Relativistic Waves and Quantum Fields Time Allowed: 2 hours 30 minutes Date: XX th May, 2010 Time: 14:30-17:00 Instructions: Answer THREE QUESTIONS only. Each question
More informationDirac Equation. Chapter 1
Chapter Dirac Equation This course will be devoted principally to an exposition of the dynamics of Abelian and non-abelian gauge theories. These form the basis of the Standard Model, that is, the theory
More informationWe will also need transformation properties of fermion bilinears:
We will also need transformation properties of fermion bilinears: Parity: some product of gamma matrices, such that so that is hermitian. we easily find: 88 And so the corresponding bilinears transform
More information2.1 Green Functions in Quantum Mechanics
Chapter 2 Green Functions and Observables 2.1 Green Functions in Quantum Mechanics We will be interested in studying the properties of the ground state of a quantum mechanical many particle system. We
More informationPRINCIPLES OF PHYSICS. \Hp. Ni Jun TSINGHUA. Physics. From Quantum Field Theory. to Classical Mechanics. World Scientific. Vol.2. Report and Review in
LONDON BEIJING HONG TSINGHUA Report and Review in Physics Vol2 PRINCIPLES OF PHYSICS From Quantum Field Theory to Classical Mechanics Ni Jun Tsinghua University, China NEW JERSEY \Hp SINGAPORE World Scientific
More informationWeek 5-6: Lectures The Charged Scalar Field
Notes for Phys. 610, 2011. These summaries are meant to be informal, and are subject to revision, elaboration and correction. They will be based on material covered in class, but may differ from it by
More informationGeometry and Physics. Amer Iqbal. March 4, 2010
March 4, 2010 Many uses of Mathematics in Physics The language of the physical world is mathematics. Quantitative understanding of the world around us requires the precise language of mathematics. Symmetries
More informationPreliminaries: what you need to know
January 7, 2014 Preliminaries: what you need to know Asaf Pe er 1 Quantum field theory (QFT) is the theoretical framework that forms the basis for the modern description of sub-atomic particles and their
More informationLECTURES ON QUANTUM MECHANICS
LECTURES ON QUANTUM MECHANICS GORDON BAYM Unitsersity of Illinois A II I' Advanced Bock Progrant A Member of the Perseus Books Group CONTENTS Preface v Chapter 1 Photon Polarization 1 Transformation of
More informationQuantum Mechanics II
Quantum Mechanics II Prof. Boris Altshuler March 8, 011 1 Lecture 19 1.1 Second Quantization Recall our results from simple harmonic oscillator. We know the Hamiltonian very well so no need to repeat here.
More informationThe Dirac Equation. H. A. Tanaka
The Dirac Equation H. A. Tanaka Relativistic Wave Equations: In non-relativistic quantum mechanics, we have the Schrödinger Equation: H = i t H = p2 2m 2 = i 2m 2 p t i Inspired by this, Klein and Gordon
More informationGeneral Relativity in a Nutshell
General Relativity in a Nutshell (And Beyond) Federico Faldino Dipartimento di Matematica Università degli Studi di Genova 27/04/2016 1 Gravity and General Relativity 2 Quantum Mechanics, Quantum Field
More informationBeta functions in quantum electrodynamics
Beta functions in quantum electrodynamics based on S-66 Let s calculate the beta function in QED: the dictionary: Note! following the usual procedure: we find: or equivalently: For a theory with N Dirac
More informationQuantization of a Scalar Field
Quantization of a Scalar Field Required reading: Zwiebach 0.-4,.4 Suggested reading: Your favorite quantum text Any quantum field theory text Quantizing a harmonic oscillator: Let s start by reviewing
More information3P1a Quantum Field Theory: Example Sheet 1 Michaelmas 2016
3P1a Quantum Field Theory: Example Sheet 1 Michaelmas 016 Corrections and suggestions should be emailed to B.C.Allanach@damtp.cam.ac.uk. Starred questions may be handed in to your supervisor for feedback
More informationVector Fields. It is standard to define F µν = µ ϕ ν ν ϕ µ, so that the action may be written compactly as
Vector Fields The most general Poincaré-invariant local quadratic action for a vector field with no more than first derivatives on the fields (ensuring that classical evolution is determined based on the
More informationIntroduction to particle physics Lecture 2
Introduction to particle physics Lecture 2 Frank Krauss IPPP Durham U Durham, Epiphany term 2009 Outline 1 Quantum field theory Relativistic quantum mechanics Merging special relativity and quantum mechanics
More informationChapter 10 Operators of the scalar Klein Gordon field. from my book: Understanding Relativistic Quantum Field Theory.
Chapter 10 Operators of the scalar Klein Gordon field from my book: Understanding Relativistic Quantum Field Theory Hans de Vries November 11, 2008 2 Chapter Contents 10 Operators of the scalar Klein Gordon
More informationParticle Notes. Ryan D. Reece
Particle Notes Ryan D. Reece July 9, 2007 Chapter 1 Preliminaries 1.1 Overview of Special Relativity 1.1.1 Lorentz Boosts Searches in the later part 19th century for the coordinate transformation that
More informationChapter 1 LORENTZ/POINCARE INVARIANCE. 1.1 The Lorentz Algebra
Chapter 1 LORENTZ/POINCARE INVARIANCE 1.1 The Lorentz Algebra The requirement of relativistic invariance on any fundamental physical system amounts to invariance under Lorentz Transformations. These transformations
More informationLecture 5: Sept. 19, 2013 First Applications of Noether s Theorem. 1 Translation Invariance. Last Latexed: September 18, 2013 at 14:24 1
Last Latexed: September 18, 2013 at 14:24 1 Lecture 5: Sept. 19, 2013 First Applications of Noether s Theorem Copyright c 2005 by Joel A. Shapiro Now it is time to use the very powerful though abstract
More informationLecture notes for QFT I (662)
Preprint typeset in JHEP style - PAPER VERSION Lecture notes for QFT I (66) Martin Kruczenski Department of Physics, Purdue University, 55 Northwestern Avenue, W. Lafayette, IN 47907-036. E-mail: markru@purdue.edu
More informationExercises Symmetries in Particle Physics
Exercises Symmetries in Particle Physics 1. A particle is moving in an external field. Which components of the momentum p and the angular momentum L are conserved? a) Field of an infinite homogeneous plane.
More informationThe Hamiltonian operator and states
The Hamiltonian operator and states March 30, 06 Calculation of the Hamiltonian operator This is our first typical quantum field theory calculation. They re a bit to keep track of, but not really that
More informationLecture 8. September 21, General plan for construction of Standard Model theory. Choice of gauge symmetries for the Standard Model
Lecture 8 September 21, 2017 Today General plan for construction of Standard Model theory Properties of SU(n) transformations (review) Choice of gauge symmetries for the Standard Model Use of Lagrangian
More informationLecture notes for FYS610 Many particle Quantum Mechanics
UNIVERSITETET I STAVANGER Institutt for matematikk og naturvitenskap Lecture notes for FYS610 Many particle Quantum Mechanics Note 20, 19.4 2017 Additions and comments to Quantum Field Theory and the Standard
More information1 Free real scalar field
1 Free real scalar field The Hamiltonian is H = d 3 xh = 1 d 3 x p(x) +( φ) + m φ Let us expand both φ and p in Fourier series: d 3 p φ(t, x) = ω(p) φ(t, x)e ip x, p(t, x) = ω(p) p(t, x)eip x. where ω(p)
More informationB = 0. E = 1 c. E = 4πρ
Photons In this section, we will treat the electromagnetic field quantum mechanically. We start by recording the Maxwell equations. As usual, we expect these equations to hold both classically and quantum
More informationParity P : x x, t t, (1.116a) Time reversal T : x x, t t. (1.116b)
4 Version of February 4, 005 CHAPTER. DIRAC EQUATION (0, 0) is a scalar. (/, 0) is a left-handed spinor. (0, /) is a right-handed spinor. (/, /) is a vector. Before discussing spinors in detail, let us
More informationPhysics 557 Lecture 5
Physics 557 Lecture 5 Group heory: Since symmetries and the use of group theory is so much a part of recent progress in particle physics we will take a small detour to introduce the basic structure (as
More informationLecture 4 - Relativistic wave equations. Relativistic wave equations must satisfy several general postulates. These are;
Lecture 4 - Relativistic wave equations Postulates Relativistic wave equations must satisfy several general postulates. These are;. The equation is developed for a field amplitude function, ψ 2. The normal
More informationPhysics 218. Quantum Field Theory. Professor Dine. Green s Functions and S Matrices from the Operator (Hamiltonian) Viewpoint
Physics 28. Quantum Field Theory. Professor Dine Green s Functions and S Matrices from the Operator (Hamiltonian) Viewpoint Field Theory in a Box Consider a real scalar field, with lagrangian L = 2 ( µφ)
More informationParticle Physics I Lecture Exam Question Sheet
Particle Physics I Lecture Exam Question Sheet Five out of these 16 questions will be given to you at the beginning of the exam. (1) (a) Which are the different fundamental interactions that exist in Nature?
More informationProblem Set No. 1: Quantization of Non-Relativistic Fermi Systems Due Date: September 14, Second Quantization of an Elastic Solid
Physics 56, Fall Semester 5 Professor Eduardo Fradkin Problem Set No. : Quantization of Non-Relativistic Fermi Systems Due Date: September 4, 5 Second Quantization of an Elastic Solid Consider a three-dimensional
More informationLorentz-covariant spectrum of single-particle states and their field theory Physics 230A, Spring 2007, Hitoshi Murayama
Lorentz-covariant spectrum of single-particle states and their field theory Physics 30A, Spring 007, Hitoshi Murayama 1 Poincaré Symmetry In order to understand the number of degrees of freedom we need
More informationAs usual, these notes are intended for use by class participants only, and are not for circulation. Week 7: Lectures 13, 14.
As usual, these notes are intended for use by class participants only, and are not for circulation. Week 7: Lectures 13, 14 Majorana spinors March 15, 2012 So far, we have only considered massless, two-component
More informationParis Samuel Miles-Brenden. June 23, 2017
Symmetry of Covariance & Exchange: Particle, Field Theory & The Two Body Equation Paris Samuel Miles-Brenden June 23, 2017 Introduction The interior and effective representational algebra of the raising
More information129 Lecture Notes More on Dirac Equation
19 Lecture Notes More on Dirac Equation 1 Ultra-relativistic Limit We have solved the Diraction in the Lecture Notes on Relativistic Quantum Mechanics, and saw that the upper lower two components are large
More information1.3 Translational Invariance
1.3. TRANSLATIONAL INVARIANCE 7 Version of January 28, 2005 Thus the required rotational invariance statement is verified: [J, H] = [L + 1 Σ, H] = iα p iα p = 0. (1.49) 2 1.3 Translational Invariance One
More informationQuantum Mechanics: Fundamentals
Kurt Gottfried Tung-Mow Yan Quantum Mechanics: Fundamentals Second Edition With 75 Figures Springer Preface vii Fundamental Concepts 1 1.1 Complementarity and Uncertainty 1 (a) Complementarity 2 (b) The
More informationIntroduction to relativistic quantum mechanics
Introduction to relativistic quantum mechanics. Tensor notation In this book, we will most often use so-called natural units, which means that we have set c = and =. Furthermore, a general 4-vector will
More informationHamiltonian Field Theory
Hamiltonian Field Theory August 31, 016 1 Introduction So far we have treated classical field theory using Lagrangian and an action principle for Lagrangian. This approach is called Lagrangian field theory
More informationClassical field theory 2012 (NS-364B) Feynman propagator
Classical field theory 212 (NS-364B Feynman propagator 1. Introduction States in quantum mechanics in Schrödinger picture evolve as ( Ψt = Û(t,t Ψt, Û(t,t = T exp ı t dt Ĥ(t, (1 t where Û(t,t denotes the
More informationQFT Perturbation Theory
QFT Perturbation Theory Ling-Fong Li (Institute) Slide_04 1 / 43 Interaction Theory As an illustration, take electromagnetic interaction. Lagrangian density is The combination L = ψ (x ) γ µ ( i µ ea µ
More informationQuantum Field Theory I Examination questions will be composed from those below and from questions in the textbook and previous exams
Quantum Field Theory I Examination questions will be composed from those below and from questions in the textbook and previous exams III. Quantization of constrained systems and Maxwell s theory 1. The
More informationRelativistic quantum mechanics
Chapter 6 Relativistic quantum mechanics The Schrödinger equation for a free particle in the coordinate representation, i Ψ t = 2 2m 2 Ψ, is manifestly not Lorentz constant since time and space derivatives
More informationG : Quantum Mechanics II
G5.666: Quantum Mechanics II Notes for Lecture 5 I. REPRESENTING STATES IN THE FULL HILBERT SPACE Given a representation of the states that span the spin Hilbert space, we now need to consider the problem
More informationAs usual, these notes are intended for use by class participants only, and are not for circulation. Week 6: Lectures 11, 12
As usual, these notes are intended for use by class participants only, and are not for circulation Week 6: Lectures, The Dirac equation and algebra March 5, 0 The Lagrange density for the Dirac equation
More informationIntroduction to Quantum Mechanics PVK - Solutions. Nicolas Lanzetti
Introduction to Quantum Mechanics PVK - Solutions Nicolas Lanzetti lnicolas@student.ethz.ch 1 Contents 1 The Wave Function and the Schrödinger Equation 3 1.1 Quick Checks......................................
More informationCoupling of Angular Momenta Isospin Nucleon-Nucleon Interaction
Lecture 5 Coupling of Angular Momenta Isospin Nucleon-Nucleon Interaction WS0/3: Introduction to Nuclear and Particle Physics,, Part I I. Angular Momentum Operator Rotation R(θ): in polar coordinates the
More informationLecture 10: A (Brief) Introduction to Group Theory (See Chapter 3.13 in Boas, 3rd Edition)
Lecture 0: A (Brief) Introduction to Group heory (See Chapter 3.3 in Boas, 3rd Edition) Having gained some new experience with matrices, which provide us with representations of groups, and because symmetries
More information1 The postulates of quantum mechanics
1 The postulates of quantum mechanics The postulates of quantum mechanics were derived after a long process of trial and error. These postulates provide a connection between the physical world and the
More information