* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Document
Bohr–Einstein debates wikipedia , lookup
Classical mechanics wikipedia , lookup
EPR paradox wikipedia , lookup
Four-vector wikipedia , lookup
Spin (physics) wikipedia , lookup
Relational approach to quantum physics wikipedia , lookup
Bell's theorem wikipedia , lookup
Yang–Mills theory wikipedia , lookup
Hydrogen atom wikipedia , lookup
Condensed matter physics wikipedia , lookup
Field (physics) wikipedia , lookup
Quantum field theory wikipedia , lookup
Quantum electrodynamics wikipedia , lookup
Quantum vacuum thruster wikipedia , lookup
Aharonov–Bohm effect wikipedia , lookup
Path integral formulation wikipedia , lookup
Renormalization wikipedia , lookup
Old quantum theory wikipedia , lookup
Elementary particle wikipedia , lookup
History of subatomic physics wikipedia , lookup
Time in physics wikipedia , lookup
Electromagnetism wikipedia , lookup
Nuclear structure wikipedia , lookup
Grand Unified Theory wikipedia , lookup
Quantum chromodynamics wikipedia , lookup
Fundamental interaction wikipedia , lookup
Standard Model wikipedia , lookup
History of quantum field theory wikipedia , lookup
Photon polarization wikipedia , lookup
Noether's theorem wikipedia , lookup
Relativistic quantum mechanics wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Chien-Shiung Wu wikipedia , lookup
Symmetry in quantum mechanics wikipedia , lookup
Introduction to gauge theory wikipedia , lookup
Mathematical formulation of the Standard Model wikipedia , lookup
Symmetries and Conservation Laws 1. Costituents of Matter 2. Fundamental Forces 3. Particle Detectors 4. Symmetries and Conservation Laws 5. Relativistic Kinematics 6. The Static Quark Model 7. The Weak Interaction 8. Introduction to the Standard Model 9. CP Violation in the Standard Model (N. Neri) “Five Deities Mandala Tibet, XVIIth Century The word is used also to indicate a circular diagram, basically made by the association of different geometric figures (the most used being the dot, the triangle, the circle and the square). The drawing has spiritual and ritual meaning in both Buddhism and Hinduism. “Mandala” from it.wikipedia.org 1 A classification of symmetries in particle physics Class Invariance Conserved quantity Proper orthochronous Lorentz symmetry translation in time (homogeneity) energy translation in space (homogeneity) linear momentum rotation in space (isotropy) angular momentum P, coordinate inversion spatial parity C, charge conjugation charge parity T, time reversal CPT time parity product of parities U(1) gauge transformation electric charge U(1) gauge transformation lepton generation number U(1) gauge transformation hypercharge U(1)Y gauge transformation weak hypercharge U(2) [U(1) × SU(2)] electroweak force Discrete symmetry Wikipedia: Internal symmetry (independent of spacetime coordinates) SU(2) gauge transformation Isospin SU(2)L gauge transformation weak isospin P × SU(2) G-parity SU(3) "winding number" baryon number SU(3) gauge transformation quark color SU(3) (approximate) quark flavor S(U(2) × U(3)) [ U(1) × SU(2) × SU(3)] Standard Model 2 Symmetries of a physical system: Classical system Lagrangian Formalism Hamiltonian Formalism Invariance of Equations of Motion Quantum system Lagrangian Formalism Hamiltonian Formalism •Invariance of dynamical equations •Invariance of commutation relations (Invariance of probability) E. Noether’s Theorem (valid for any lagrangian theory, classical or quantum) relates symmetries to conserved quantities of a physical system 3 A “classical” example : T 1 2 1 m1r1 m2 r22 2 2 r1 r2 V V (r1 r2 ) m1r1 V (r1 r2 ) r1 m2 r2 V (r1 r2 ) r2 ' Let us do a translation : ri ri ri a V (r1 r2 ) V (r1 a r2 a) V (r1 r2 ) ' mi ri ' V (r1 r2 ) ri The equations of motion are translation Invariant ! 4 If one calculates the forces acting on 1 and 2: FTOT F1 F2 V (r1 r2 ) V (r1 r2 ) V (r1 r2 ) V (r1 r2 ) 0 r1 r2 r2 r2 dPTOT FTOT 0 dt In the classical Lagrangian formalism : L L(qi , qi ) d L L 0 dt qi qi L pi qi L invariant with respect to q dpi L dt qi p conserved 5 In the Hamiltonian formalism qi qi , H p i pi , H d (qi , pi ) , H dt Possible conservation of a dinamical quantity Commutation (Poisson brackets) with the Hamiltonian conservation of dynamical quantities Possible symmetry This formalism can easily be extended to Quantum Mechanics In Quantum Mechanics, starting from the Schroedinger Equation : i s (t ) H S (t ) t s (t ) exp i(t t0 ) H / S (t0 ) T (t , t0 ) Time evolution (unitary operator) 6 Schroedinger and Heisenberg Pictures: Heisenberg Schroedinger Q s (t0 )* Q(t ) S (t0 ) dV S (t )* Q0 S (t ) dV S (t0 )* Q(t ) S (t0 ) S (t )* Q0 S (t ) S (t0 )* Q(t ) S (t0 ) S (t0 )* T 1Q0 T S (t0 ) Q(t ) T 1Q0 T for the operators in the Heisenberg picture Taking the derivatives: d dT 1 dT i Q(t ) i Q0 T iT 1Q0 HT 1Q0T T 1Q0TH dt dt dt i d Q(t ) HQ QH Q, H dt d Q i Q(t ) i Q, H dt t Conserved quantities: commute with H In the case when there is an explicit time dependence (non-isolated systems) 7 Translational invariance: a continuous spacetime symmetry (r r ) (r ) r 1 r D r r i D ( r ) 1 p r The translation operator is naturally associated to the linear momentum For a finite translation : i i r i D (r ) lim 1 p r lim 1 p exp p r n n n n unitary n ( r n r ) Self-adjoint: the generator of space translations If H does not depend on coordinates D, H 0 p, H 0 The momentum is conserved 8 Rotational invariance: a continuous spacetime symmetry i 1 J z R () 1 The rotation operator is naturally associated with the angular momentum J z i x y i x y Angular momentum (z-component) operator (angle phi) Self-adjoint: rotation generator A finite rotation unitary n i i R ( ) lim 1 J z exp J z n n If H does not depend on the rotation angle φ around the z-axis R, H 0 J z , H 0 The angular momentum is conserved 9 Time invariance (a continuous symmetry) The generator of time translation is actually the energy! s (t ) exp i(t t0 ) H / S (t0 ) Using the equation of motion of the operators : i d Q H (t ) i Q, H dt t i d H H (t ) i H , H dt t If H does not dipend from t, the energy is conserved The continous spacetime symmetries: Space translation Space rotation Time translation Linear momenum Angular momentum Energy 10 Continuous symmetries and groups: the case of SU(2) Combination of two transformations: the result depends on the commutation rules of the group generators. For instance, in the case of space translations : pi , p j 0 Commutative (Abelian) algebra of translations Translation operator along x: i Dx ( ) exp p x i i i i Di ( ) D j ( ) exp pi exp p j exp p j exp pi D j ( ) Di ( ) (two translations commute). Moreover: i i i Dx ( ) Dx ( ) exp p x exp p x exp p j ( ) Dx ( ) and clearly : i Dx (0) exp p x 0 1 11 In the case of rotations : Commutation rules for the generators: i Rz ( ) exp Lz L , L i j k jkl Ll A non-commutative algebra Rotations about different axes do not commute This is also the case of SU(2), which we are now going to introduce Why SU(2) ? In the case of a two level quantum system, the relevant internal symmetries are described by the SU(2) group, having algebraic structure similar to O(3). SU(2) finds an application in the Electroweak Theory. SU(3) can instead be applied to QCD, the candidate theory of Strong Interactions. 12 Isospin Symmetry, SU(2) Let us consider a two-state quantum system (the original idea of this came from the neutron and the proton, considered to be degenerate states of the nuclear force). Since they were considered degenerate, they could be redefined : H p E p p 'p p n H 'p E 'p H n E n n n' p n H n' E n' Degeneration Redefinition Double degeneracy, similar to what happens in s=1/2 spin systems. The degeneration can be removed by a magnetic field One can introduce a two-components spinor : (1/ 2 ) p p 1p/ 2 n n1/ 2 n 1/ 2 p 1 0 1/ 2 n 0 1 13 (1/ 2) (1/ 2) ' U (1/ 2) The ridefinition now becomes : A symmetry for the Strong Interactions (broken by electromagnetism) U U 1 det U 1 U 1 i SU(2) is a Lie group Properties can be deduced from infinitesimal transformations (1 i )(1 i ) 1 Tr 0 det U 1 Which can be written in a general form : 2 Pauli matrices 0 1 1 1 0 0 i 2 i 0 1 0 3 0 1 i j k , i ijk 2 2 2 14 1 S 2 (p1/ 2) 2 (p1/ 2) s ( s 1) (p1/ 2 ) 4 1 1 S z (p1/ 2 ) 3 (p1/ 2) s z (p1/ 2) (p1/ 2 ) 2 2 1 S 2 n(1/ 2) 2 n(1/ 2) s ( s 1) n(1/ 2 ) 4 1 1 (1/ 2) S z n(1/ 2 ) 3 n(1/ 2 ) s z n(1/ 2 ) p 2 2 'p 1 i p ' 2 n n Infinitesimal rotation of the p-n doublet : A finite rotation in SU(2): U lim 1 i exp i / 2 n n 2 n • Generalization of a global phase transformation • Three phase angles • Non-commuting operators (Non-abelian phase invariance) example (1/ 2 ) ' exp i / 2 (1/ 2) 'p cos 2 ' n sin 2 sin 2 p n cos 2 15 The two nucleon system 1 1p/ 2 0 A one-nucleon state can be described with the base of the nuclear spinors 0 n1/ 2 1 A two nucleon state can be constructed by combining the Isospin states : N ) I 1 / 2, I 3 1 / 2, 1 / 2 n 1/ 2 1/ 2 I , I3 : N ) I 1 / 2, I 3 1 / 2, 1 / 2 p n I 1 I3 1 0 1 I 0 I3 0 1,1 , 1,0 , 1,1 p 0,0 16 1,1 pp Isospin triplet 1 pn np 1,0 2 1,1 nn 1 pn np 2 0,0 In an Isospin rotation: These states transform into one another in Isospin rotations, similar to a 3-d vector for ordinary rotations Isospin singlet (scalar) 0,0 0,0 Isospin Invariance means: there are two amplitudes, I=0 e I=1 I=1 states cannot be internally distinguished by the Strong Interactions Strong Interactions conserves I and I3 (full I-spin invariance) Electromagnetic interaction conserves I but not I3 (the charge is different in different states of the same I-spin multiplet, e.g. the proton and neutron) 17 Isospin invariance at work: pion-nucleon reactions N N Let us consider the (strong) diffusion of pions and nucleons. The diffusion amplitudes are the element of the S matrix: , m ' ; N , l ' S , m; N , l which can be written as combinations of the total isospin of the system : , N ; I ' , I 3' S , N ; I , I 3 II I I S ( I , I 3 ) ' ' 3 3 taking into account the conservation of I and I3 Note, however, that S must commute also with I1 and I2 which implies (Schur’s Lemma) that S is S=(I). Therefore, the amplitude can be expressed as a function of the allowed total isospin amplitudes. S ( I 1 / 2) S ( I 3 / 2) 18 ) I 1, I 3 1, 0, 1 N ) I 1 / 2, I 3 1 / 2, 1 / 2 0 n I 1/ 2 11/ 2 I 3/ 2 I3 p 1/ 2 1/ 2 I3 3 / 2 1/ 2 1/ 2 3 / 2 Using the two invariant amplitudes (1/2 and 3/2), one can write the amplitudes of the following processes : p 3 / 2,3 / 2 p 0p 2 3 / 2,1 / 2 3 1 1 / 2,1 / 2 3 p p n 1 3 / 2,1 / 2 3 2 1 / 2,1 / 2 3 p 1 3 / 2,1 / 2 3 2 1 / 2,1 / 2 3 0n 2 3 / 2,1 / 2 3 1 1 / 2,1 / 2 3 p p 0 n 19 p 3 / 2,3 / 2 0p 2 3 / 2,1 / 2 3 1 1 / 2,1 / 2 3 n 1 3 / 2,1 / 2 3 2 1 / 2,1 / 2 3 p 1 3 / 2,1 / 2 3 2 1 / 2,1 / 2 3 2 3 / 2,1 / 2 3 1 1 / 2,1 / 2 3 0n Some Isospin amplitudes that one can study by making Pion-Nucleon diffusion experiments π π N N A( p p) 3 / 2,3 / 2 S 3 / 2,3 / 2 S (3 / 2) 1 2 1 2 3 / 2,1 / 2 S 3 / 2,1 / 2 1 / 2,1 / 2 S 1 / 2,1 / 2 S (3 / 2) S (1 / 2) 3 3 3 3 2 2 A( p 0 n) S (3 / 2) S (1 / 2) 3 3 A( p p) Let us make the Pion-Nucleon experiment at an energy of the Pion of about 200 MeV. This generates a clear resonant peak in the π+p cross section, indicating the presence of a dominant I=3/2 resonance (the Δ++, m = 1232 MeV) 20 At the energy when the cross section resonates with the Δ++ state, the 3/2 amplitude is dominant. Therefore S(1/2) can be neglected at that energy and we have a prediction of relative cross sections of : ( p p) p π+ cross section dominates at the energy of the Delta resonance √s GeV A( p p) S (3 / 2) A 1 1 2 1 A( p p) S (3 / 2) S (1 / 2) S (3 / 2) 3 3 3 2 2 2 A( p 0 n) S (3 / 2) S (1 / 2) S (3 / 2) 3 3 3 A 2 1 9 2 2 A 9 2 In agreement with the experimental observation of ( p p) 3 p p p 0 n 21 A little preview of a Lagrangian for a system of particles: Nucleon, Pions and some Interaction (Yukawa style) between them. This effective Lagrangian could describe reasonably well hadronic physics at energies below ~2 GeV (assuming the proton and the pion to be elementary) 1 L N i M N i i i2 mi2 g i N i i 5 N i 2 Will produce the Dirac Equation A free non interacting Nucleon (Kinetic Energy and Mass) Will produce the Klein-Gordon Equation. This term represents free Pions (Kinetic Energy and Mass terms for each Pion) A Yukawa interaction term (interaction term in the simplest form, taking into account pion parity through the Gamma-5 matrix) 22 Nucleons and Quarks I3 particle antiparticle particle +1/2 p -1/2 n n p u d antiparticle d u An Isospin triplet : the Pion Wave Function I 1 1 I3 Q/e 1 ud 1 -1 u d -1 1 0 0 o 1 uu dd 0 2 1 dd uu 2 o 0 23 Building up strongly interacting particles (hadrons) using Quarks as building blocks 24 Will need to reconsider later on (Static Quark Model section) 25 (1,1) p (1) p (2) More on the two nucleon state: S Isospin part The total wave function : A (tot) ( space) ( spin ) (isospin ) 1 p(1)n(2) n(1) p(2) 2 (1,1) n(1)n(2) 1 p(1)n(2) n(1) p(2) (0,0) 2 (1,0) (non relativistic decomposition) The Deuton case: As it is known the deuton spin = 1 α symmetric φ has (-)l symm. It is known that the two nucleons are l=0 or l=2, φ is symm. Then must be antisymmetric. This is because ψ tot must be antisymmetric in nucleon exchange. I 0 The Deuton is an Isospin singlet Now, since the Deuton has I=0 and the Pion has I=1, considering the reactions : (i) p p d Isospin I 1 1 The reaction can proceed only via I=1 (ii ) p n d 0 0,1 1 ( pn d 0 ) / ( pp d ) 1 / 2 26 Gauge symmetries (global and local) Gauge symmetries are continuous symmetries (a continuous symmetry group). They can be global or local. Global symmetries: conserved quantities (electric charge) Local symmetries: new fields and their transformation laws (Gauge theories) Let us consider the Schoedinger equation 2 2 V (r ) (r ) E (r ) 2m Let us consider a global phase transformation: the change in phase is the same everywhere i (r ) e (r ) The Schroedinger equation is invariant for this transformation. This invariance ia associated (E. Noether’s Theorem) to electric charge conservation. But then what happens if we consider a local gauge tranformation ? k (r ) 27 i (r ) e (r ) i ( r ) (r ) e (r ) How does one realize a local gauge invariance ? i ( r ) (r ) e (r ) 2 2 i ( r ) i ( r ) V (r ) e (r ) E e (r ) 2m Non invariant! And this is because : i ( r ) i i ( r ) e (r ) e i e (r ) extra term ! 28 To solve the problem (and stick to the invariance) we can introduce a new field. The field would compensate for the extra term. The new field would need to have an appropriate transformation law. Since the free Schroedinger Equation is not invariant under: 2 i ( x, t ) i ( x, t ) 2m t e ' Let us modify the Equation : Compensating fields Transformation laws: i q ( x ,t ) 2 i G ( x , t ) i R ( x , t ) 2m t ' G G G q R R ' R iq t 29 This allows us to restore the invariance Gq A R iqV To give the fields physical meaning : It is invariant, indeed iq e ' 2 i qA ( x, t ) i iqV ( x, t ) 2m t ' A A A V V ' V t ' i qA 2m ( x, t ) i iqV 2 ' t ' ' ( x , t ) The local gauge U(1) invariance of the free Schroedinger field • Requires the presence of the Electromagnetic Field • Dictates the field transformation law 30 This gauge symmetry is the U(1) gauge symmetry related to phase invariance: ' e i q ( x ,t ) There are, of course, other possibilities. In physics, a gauge principle specifies a procedure for obtaining an interaction term from a free Lagrangian which is symmetric with respect to a continuous symmetry -- the results of localizing (or gauging) the global symmetry group must be accompanied by the inclusion of additional fields (such as the electromagnetic field), with appropriate kinetic and interaction terms in the action, in such a way that the extended Lagrangian is covariant with respect to a new extended group of local transformations. FREE INTERACTING 2 i ( x, t ) i ( x , t ) 2m t 2 i qA ( x, t ) i iqV ( x, t ) 2m t 31 The U(1) Gauge Invariance and the Dirac field Dirac Equation (1928): a quantum-mechanical description of spin ½ elementary particles, compatible with Special Relativity i i m t 0 i i i 0 i m 0 4 component spinor ( x) ( x) 0 Conjugate spinor 1 0 0 1 0 Matrices 4x4 (gamma) Dirac probability current j j 0 32 L i c mc 2 The Dirac Lagrangian: Features a global gauge invariance: ei (phase transformations) Let us require this global invariance to hold locally. The invariance is now a dynamical principle i ( x ) e e iq ( x ) c Now the gauge transformation depends on the spacetime points Let us see how L behaves : Since iq iq iqc iq e e c e c c 33 L L' i c e iq c iq c e mc 2 L q This lagrangian is NOT gauge-invariant. If we want a gauge-invariant L, one has to introduce a compensating field with a suitable transformation law : L i c mc 2 (q ) A A A This new lagrangian is locally gauge-invariant. This was made possible by the introduction of a new field (the E.M. field). 34 L i c mc 2 (q ) A ei ( x ) e iq ( x ) c A A iq c iqc L i c e e mc 2 (q ) ( A ) i ce iq c e iq c q mc 2 (q ) A q L The gauge field A must however include a (gauge-invariant) free-field term. This will be the E.M. free field term (more on this in the Standard Model lecture) 1 L ic ( mc ) (q ) A F F 16 2 35 Discrete symmetries: P,C,T Discrete symmetries describe non-continuous changes in a system. They cannot be obtained by integrating infinitesimal transformations. These transformations are associated to discrete symmetry groups Parity P Inversion of all space coordinates : x x P: y y z z P (r ) (r ) The determinant of this transform is -1. In the case of rotations, that would be +1 PP (r ) P (r ) (r ) PP 1 P P PP 1 A unitary operator. Eigenvalues: +1, -1 (if definite-parity states) Eigenstates: definite parity states 36 H , P 0 Parity is conserved in a system when The case of the central potential: PH (r ) H (r ) H (r ) H (r ) Bound states of a system with radial symmetry have definite parity Example: the hydrogen atom Hydrogen atom: wavefunction (no spin) (r , , ) (r ) Yl m ( , ) Radial part The effect of parity on the state is : P : Angular part P : Yl m ( , ) (1)l Yl m ( , ) Electric Dipole Transition ∆l = ± 1. This implies a change of parity. Since, however, e.m. interactions conserve parity, this indicates the need to attribute to the photon an intrinsic (negative) parity. P ( ) 1 37 The general parity of a quantum state P ( x,t ) Pa ( x,t ) Let us consider a single particle a. The intrinsic parity can be represented by a phase P ( x,t ) exp i / 2 ( x,t ) Intrinsic Spatial P P ( x ,t ) ( x ,t ) Which is the physical meaning of the intrinsic parity ? For instance, in a plane wave (momentum eigenstates) representation P p ( x ,t ) P exp i ( px Et ) Pa p ( x ,t ) Pa p ( x ,t ) If one then lets go the momentum to zero, one can see that the intrinsic parity has the meaning of a parity in the p=0 system 38 The parity of the photon from a classical analogy A classical E field obeys : Let us take the P: To keep the Poisson Equation invariant, we need to have the following law for E : E ( x ,t ) ( x ,t ) () PE ( x ,t ) ( x ,t ) PE ( x ,t ) E ( x ,t ) And the parity operation would give : A A E ( x ,t ) t t A( x ,t ) N (k )exp i (k x Et ) A( x,t ) P A( x,t ) In order to make it consistent with the electric field transformation : P 1 On the other hand, in vacuum (no charges): 39 The action of parity on relevant physical quantities Position: P: r r Momentum: Angular Momentum: P: t t dr dr P: pm p dt dt P : L r p (r ) ( p) L Time: Charge: Current: P: E field: P: E B field: P: B k Spin: P: q q J v v J r r kq 3 kq 3 E r r s r ( s ) (r ) I k I B 2 2 r r P: 40 Symmetry at work: the parity of the Pion Parity of a complex system: overall parity times the product of the intrinsic parities of the parts of the system. Let us illustrate this for the Pion, using the reaction: d n n This is a Strong Interaction process Parity is conserved L( d ) 0 (known ) P(d ) 1 (known ) P( d ) (1) L P( ) P(d ) (1)0 P( )1 P( ) P(nn) (1) L P(n) P(n) (1) L P( ) (1) L ( L of the final state ) P conservation s 0, sd 1 s1 1 / 2, s2 1 / 2 (already known) J L S 1 initial state d n n P( ) (1)1 1 ( L of the final state ) L S 1 final state J conservation (1) LS 1 1 L 1, S 1 Pauli symmetry between neutrons L S even with J 1 41 Final considerations on parity : Some intrinsic parities cannot be observed. For instance (p, n). They are conventionally chosen to be +1. Because of the Baryon Number conservation the actual P value is not important as it cancels out in any reaction. Neutral Pion Parity. It is deduced from the photon pair distribution in : P( 0 ) P(space) (1) (1) P (space) 1 0 (study of the relative polarization of the two photons) Particles can be (and are) classified using J and P (and C) Transformation properties for rotations and space reflections. Spin-parity J P 0 : pseudoscal ar ( ) J P 0 : scalar J P 1 : vector J P 1 : axial vector The intrinsic parities of Particles and Antiparticles P(particle) = - P (antiparticle) FERMIONS P(particle) = P (antiparticle) BOSONS 42 Time Inversion T : preliminary considerations Time is a parameter characterizing the evolution of a system Classical Physics An absolute Cosmic Flow : just one Clock for the whole Universe Special and General Relativity The time flow depends on : 1. The motional status of the clock 2. The spacetime structure Is there a Time symmetry? Is there a possible Arrow of Time based on asymmetry between past and future? (A. Eddington , 1928) 43 How can we define a time symmetry ? What do we really need to run a movie backward? In Classical Physics : State A Dynamical Law State B State - A Dynamical Law State - B 1. The Initial and Final Conditions can be inverted (State - State) 2. The Dynamical Law is time-neutral : t -t leaves the law unchanged Here is a time-neutral dynamical law: d2 V ( x (t )) m 2 x(t ) dt x t t d2 V ( x (t )) m 2 x (t ) dt x d2 V Tx (t ) m 2 Tx(t ) dt x 44 Time Arrow in the MACROSCOPIC world Conventional wisdom says that physical processes at the MICROSCOPIC level are timesymmetric if the time arrow is changed in sign, the theoretical description would not change. On the other hand, in the MACROSCOPIC world, one can appreciate the existence of a preferential direction of time : from the past to the future. Building up the Arrow of Time using the Second Law of Thermodynamic. Entropy can only increase. Ink dissolved in water in the left hand side of the vessel. One has a «trivial» time sequence corresponding to an increase of Entropy. A B C D The only possible sequence is ABCD ,even if – from the MICROSCOPIC viewpoint – the opposite sequence is possible as well (and does not violate any other physical law). There is no conflict between the MICRO (T-reversible) laws and the MACRO (Tirreversibile) behaviour. The key point is that the macrostate D corresponds to a much bigger number of microstates with respect to A 45 What is the origin of the MACROSCOPIC Time Arrow? Boltzmann’s suggestion: thermodynamic time can arise from cosmic time. If the Universe had begun from an equilibrium condition, then (possibly) we would not have an Entropy concept as we know it. A B Analogy C D Since all started from a relatively ordered state, a very peculiar state indeed (the first primi istanti del Big Bang), it is then natural (emergent) a tendency to maximum Entropy it is naturally emergent a concept of macroscopic cosmic time 46 Let us suppose a different Demiurgical Creation If the Universe had begun from an equilibrium condition (see below) then (likely) we would not have a concept of Entropy like this. Likely, this means no macroscopic time. Creazione di Adamo – Michelangelo Buonarroti (Musei Vaticani - La Cappella Sistina) Two species of perfect gases (the blue and the red one). They do not interact except for perfectly elastic scatterings. Time arrow undefined 47 Time Inversion T in Particle Physics It changes the time arrow T Classical dynamical equations are invariant because of second order in time t t r r dr dr p m m p dt dt L r p r ( p) L Classical microscopic systems : T invariance is fully respected. Classical macroscopic systems: time arrow selected statistically (defined as the direction of non decrease of entropy) In the quantum case, the Schroedinger equation : Is not invariant for i (t ) H (t ) t T? (r , t ) (r ,t ) T i H t i (t ) H (t ) t 48 Let us now start from the Conjugate Schroedinger Equation : i (r , t ) H (r , t ) t i * (r , t ) H * (r , t ) t T * (r , t ) (r ,t ) And define a T-inversion operator : i * (r , t ) H * (r , t ) t T So, with this definition of T operator, we have: * * i (r ,t ) H (r ,t ) t i T (r ,t ) H T (r ,t ) t The operator representing T is an antilinear operator. The square modulus of transition amplitudes is conserved 49 i (t ) H (t ) t * i (t ) H * (t ) t T i Let us take the complex conjugate And we have the required result in terms of the conjugate wave function : i (t ) H (t ) t T * (t ) H T * (t ) t Wigner Theorem on Quantum Systems Any symmetry of a quantum system is given by: • ' U U U 1 U a b a U b U either a unitary • or an antiunitary operator ' W W W 1 W a b a*W b*W 50 x , p i The T operator in Quantum Mechanics Action on position and momentum xi T T xi pi T T pi i T xi T xi T pi T pi j ij T xi xi T T pi pi T Then considering: xi , pi i and calculating T xi , pi T T i T T xi piT T pi xi T T i T xi T piT pi T xi T T i T xi pi pi xi T i T The operator has to be antilinear i T i T 51 Non-interacting particles in the initial state Particle Physics : what do we really measure ? p1 Quantum interference. Unitary evolution. Non-interacting particles in the final state p3 p2 S p4 Performing a momentum measurement of a final states involves (in general) a selection process that traces away the other quantum mechanical degrees of freedom (exception: Dalitz Plots). p x p vt p c 200 MeV fm 200 MeV fm 200 MeV fm p pvt pvct E vt E c t E 3 10 23 ( fm / s) t A measurement on a particle can be done during a relatively long time a momentum eigenstate can be built with (almost) arbitrary accuracy : p MeV s MeV ns 10 21 10 12 p E t E t 52 We see the consequence of T-invariance at the microscopic level by analizing the the transition amplitudes when no interference is present between i and f. One single transition amplitude has to be involved. Otherwise the measurement process would generate phase cancellation and irreversibility ! The selection is made on momentum eigenstate of the final state. If these conditions are met, one can use the detailed balance principle on the initial and final momentum eigenstates : M i f M f i Note: the detailed balance DOES NOT imply the equality of the reaction rates: M i f M f i Wi f 2 2 2 2 M i f f M f i i W f i A “classical” test, the study of the (Strong) reaction p 27Al 24Mg T is violated at the microscopic level il the Weak Nuclear Interactions Physical Review Letters 109 (2012) 211801. BaBar experiment at SLAC 0 Comparing the reactions: B B 53 Charge Conjugation C C An internal discrete symmetry q q It changes the sign of the charges (and magnetic moments) r r C : E k q 3 k (q) 3 E r r s r s r C : B k 2 I k 2 ( I ) B r r In the case of a quantum state C (q, r , t ) (q, r , t ) C C The C eigenstates are the neutral states C ( ) 1 For the photon case 54 The charge conjugation of the photon from a classical analogy A classical E field obeys : Let us take the C: E ( x ,t ) ( x ,t ) CE ( x ,t ) C ( x ,t ) ( x ,t ) To keep the Poisson Equation invariant, we need to have the following law for E : On the other hand, for a system of charges : A E ( x ,t ) t CE ( x ,t ) E ( x ,t ) And the charge conjugation operation would give : ( x ,t ) C ( x ,t ) In order to make it consistent with the electric field transformation : C 1 55 The C-parity reverses the signs of charges and magnetic moments. The electromagnetic interaction is not affected. For interactions that are C-invariant : C, H 0 a a Let us distinguish between particles that have antiparticles : And particles that don’t : a , K ,e , 0 ,... The action of the C operator : Spin, momentum Mass, type C a, a , C , C , C phase C C 1 CC 1 C 1 56 The C-parity of a state can be calculated for a neutral state if we know the wave function of the state. It is the product between the parities of the total wavefunction and the parities of the constituents. Since charge conjugation of two particles of opposite charge, swaps the identify of the particles, one has to account for the proper quantum statistics C , L (1) L True also in general for spin zero particles For a couple of femions, instead : In the pi-zero decay This decay in 3 photons 0 C f f , L, S (1) L S f f C ( 0 ) C ( )C ( ) (1) (1) 1 0 Is forbidden if C is conserved in electromagnetic interactions. In fact : 0 7 10 0 57 Action of C,P,T r t mdr p dt Lr p q C r t p L P r t r ( p) L q ( v ) J q(r ) E r3 (s ) (r ) 2 I B qr E r3 (s ) r r 2 p L q J E q ( v ) J r r t mdr p d ( t ) (r ) ( p) L CPT r t md ( r ) p dt q J v v J qr ( q ) r E E 3 r3 r s r s r B 2 I ( I ) B 2 r r T I B B 58 Positronium Similar to the H atom. Actually, the «true» atom. We require the total wavefunction to be antysimmetric, considering the electron and the positron as different C-states of the same particle ( space) ( spin ) (C ) The space part Yml ( , ) The spin part C (1) S 1 C (1)l (1,1) 1 (1 / 2) 2 (1 / 2) Triplet Singlet The C conjugation for the Ps state : 1 1 (1 / 2) 2 (1 / 2) 1 (1 / 2) 2 (1 / 2) 2 (1,1) 1 (1 / 2) 2 (1 / 2) (1,0) (0,0) 1 1 (1 / 2) 2 (1 / 2) 1 (1 / 2) 2 (1 / 2) 2 K (1)l (1) S 1 C 59 K (1)l (1) S 1 C Positronium in the l=0 (fundamental) state Let us now calculate C ee 2 J 0, l 0, S 0 C 1 This state has C=+1 since in decays in 2 γ’s e e 3 J 1, l 0, S 1 C 1 This state has C= - 1 since in decays in 3 γ’s Singlet: K (1)l (1) S 1 C (1)0 (1)1 (1) 1 Triplet: K (1) (1) l S 1 C (1)0 (1) 2 (1) 1 Antisymmetry by electron/positron exchange C-parity conservation determines the Ps decay modes : Singlet: (2 ) 1.252 1010 s Triplet: (3 ) 1.374 107 s See the vertices in the relevant Feynman diagrams 60 Photons, Spin, Helicity B A 1 A E c t Gauge - invariant A A 1 c t Coulomb Gauge A 0 Free propagation: 2 1 A A 2 2 0 c t A e A0 exp i(kr t ) 2 A 0 e k 0 For instance: e e 1 2 x 2 y Plane wave solution Transversality condition in the Coulomb Gauge k (0,0, k ) 0 Plane polarization Ax ex A0 exp i (kz t ) Ay e y A0 exp i (kz t ) / 2 ex e y Circular polarization 61 Circular polarization /2 1 eR (ex ie y ) 2 1 eL (ex ie y ) 2 ex e y Which can be expressed by using the rotating vectors: The polarization vectors can be associated to the photon spin states : If the wave propagates along z: Lz xpy ypx 0 Let us make a rotation around the z axis : ex' ex cos e y sin e 'y ex sin e y cos ez' ez 1 ' (ex ie 'y ) 2 1 ' eL' (ex ie 'y ) 2 ez' eR' Transversality: the Jz=0 state does not exist: Photons with Jz =0 are virtual (longitudinal photons). They have: m≠0 Jz only due to spin R exp (i J z ) 1 (ex ie y ) exp( i ) 2 1 (ex ie y ) exp( i ) 2 1 Eigenstates of Jz with eigenvalues 1 +1,-1,0 0 e e k J z 1 R k J z 1 L 62 Helicity Projection of the spin in the momentum direction right handed H 1 left handed H 1 scalar H0 p E An approximate quantum number for massive particles So much better inasmuch the particle is relativistic Exact for photons The advantages of a description p , over a description p, m : • The Helicity is unchanged by rotation • • Since p J p , the helicity can be defined in a relativistic context Invariance laws in action: An example: E.M. interactions conserve Parity One can then build a Parity-violating quantity, like: P : p ( p) p Then, in E.M. interactions this quantity must be zero. And one can test this! In E.M. Interactions right-handed and left-handed photons appear with equal amplitudes. In this way they compensate to the result and conserve Parity. 63 The Neutrino C, P are violated in Weak Nuclear Interactions Neutrinos takes part only in Weak Nuclear Interactions J z 1/ 2, 1/ 2 In the massless neutrino approximation: Experimental evidence indicates that in Weak Interactions : Neutrinos are always left-handed. Antineutrinos are always right-handed ! p P R L C p L p CP R p To a very good approximation, Weak interactions conserve CP (not C, not P) 64 The CPT Theorem In a local, Lorentz-invariant quantum field theory, the interaction (Hamiltonian) is invariant with respect to the combined action of C,P,T (Pauli, Luders, Villars, 1957) A few consequences : 1) Mass of the particle = Mass of the antiparticle 2) (Magnetic moment of the particle) = -- (Magnetic moment of antiparticle) 3) Lifetime of particle = Lifetime of antiparticle Proton Protons, electrons Antiproton Electron Positron Q +e -e -e +e B o L(e) +1 -1 +1 -1 μ 2.79 (e / 2Mc ) 2.79 (e / 2 Mc ) e / 2mc e / 2mc σ /2 /2 /2 /2 65 CPT Theorem (wikipedia) In quantum field theory the CPT theorem states that any canonical (that is, local and Lorentz-covariant) quantum field theory is invariant under the CPT operation, which is a combination of three discrete transformations: charge conjugation C, parity transformation P, and time reversal T. It was first proved by G.Lüders, W.Pauli and J.Bell in the framework of Lagrangian field theory. At present, CPT is the sole combination of C, P, T observed as an exact symmetry of nature at the fundamental level. CPT and the Fundamental Interactions Concern systems for which a quantum field theory has been developed (i.e. systems subject to Strong, Electromagnetic and Weak Interactions). The relative Lagrangian is CPT-invariant. Note CPT is a flat-spacetime theorem (it does not concern Gravity) 66 Invariance of Physical Laws and Invariance of Physical Systems Let us consider the quantity E L L It is an Electric Dipole Moment (EDM) How this quantity P,T transforms ? T : E E E P : E ( E ) E Now let us consider a system bound by a P conserving interaction, like the Electromagnetic Interaction. Since P is conserved, one can expect that this quantity should be zero. H N We will consider the Ammonia Molecule : d 51030 C m H H 67 H N This system of minimum energy does not display the full symmetry of the Electromagnetic Interaction H H There are in fact two degenerate states : The full ammonia wavefunction has the form : 1 2 1 2 Which displays the symmetry of the underlying interaction. This is an example of a Broken Symmetry The minimum energy configuration does not display the symmetries of the theory Other examples : • A ferromagnet (when T is below the Curie temperature) • The Higgs potential 68 A recipe to search for violation of fundamental laws Consider a system S that obeys some interaction I. Build up a quantity Q that is violated by said interaction. Q is violated by I. Find a (non-degenerate) s state of the system S. Check that Q is zero on the state s. Example : the neutron Electric Dipole Moment (EDM) The magnetic moment (like a spin) changes sign with the T inversion The EDM changes sign with the P inversion Current best limit on neutron EDM: |dn| < 10–32e·cm "The Neutron EDM in the SM : A Review". arXiv:hep-ph/0008248 69 Particle Numbers: baryonic, flavor, and leptonic Flavor : The flavor is the quark content of a hadron Massa (MeV) Quark U D S C B p 938 uud +2 +1 0 0 0 n 940 udd +1 +2 0 0 0 Λ 1116 uds +1 +1 -1 0 0 Λc 2285 udc +1 +1 0 +1 0 π+ 140 u-dbar +1 -1 0 0 0 K- 494 s-ubar -1 0 -1 0 0 D- 1869 d-cbar 0 +1 0 -1 0 Ds+ 1970 c-sbar 0 0 +1 +1 0 B- 5279 b-ubar -1 0 0 0 -1 Υ 9460 b-bbar 0 0 0 0 0 70 Favor quantum numbers refer to quark content of hadrons They are conserved in Strong and Electromagnetic Interactions They are violated in Weak Interaction U N (u) N (u ) D N (d ) N (d ) S N (s) N (s ) Strangeness C N (c) N (c ) B N (b) N (b ) Charm T N (t ) N (t ) Beauty In a Stong Nuclear (or E.M.) process, all flavors are conserved: In Weak Interactions instead : Top p p n p (uud ) (uud ) (udd ) (uud ) (ud ) n p e e (udd ) (uud ) Baryon Number: 1 B U D S C B T 3 71 Baryon Number 1 B U D S C B T 3 The Baryon Number is equivalent to : 1 B (nQ nQ ) 3 Baryons have B=1 while Antibaryons have B = -1 Mesons have B = 0 This law follows from the conservation of the Quark Number. Quarks transform into each other. They disappear (or appear) in pairs. Flavor quantum numbers refer to the identity of quarks : Violated in Weak Interactions (Isospin: +1/2 o -1/2 in doublets) Strangeness: -1 for the s quark Charm: +1 for the c quark Bottom: -1 for the b quark Top: +1 for the t quark 72 The Leptonic Numbers: Ne N (e ) N (e ) N ( e ) N ( e ) Electronic Lepton Number N N ( ) N ( ) N ( ) N ( ) Muonic Lepton Number N N ( ) N ( ) N ( ) N ( ) Numero leptonico tauonico The Leptonic Numbers are conserved in any known interaction WITH THE EXCEPTION OF Neutrino Oscillations. In Neutrino Oscillations, they are violated. This process does not involve neutrino oscillations : e So, it does not take place . However, we can define a total lepton number : N l N e N N To the best of our knowledge the Total Lepton Number (sum of the three leptonic numbers) is conserved in every interaction. 73 74