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# Particle Physics PHYSICS 129

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This 15 page Class Notes was uploaded by Marjorie Hahn on Thursday October 22, 2015. The Class Notes belongs to PHYSICS 129 at University of California - Berkeley taught by Staff in Fall. Since its upload, it has received 16 views. For similar materials see /class/226698/physics-129-university-of-california-berkeley in Physics 2 at University of California - Berkeley.

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Date Created: 10/22/15

129A Lecture Notes Notes on Special Relativity 1 Why Relativity Particle Physics aims to study structure of space7 time and matter at its most fundamental level It necessarily means that we study physics at the shortest distance scales as possible To probe short distances7 Heisenberg7s uncertainty principle AzAp 2 712 states that we need to provide large mo mentum In other words7 we are looking at scattering phenomena of particles with large momenta and hence high energy In practice7 it often means the use of particle accelerators that can bombard projectile particles beams on target or opposing beams at very high momenta or energies This is why terms particle physics77 and high energy physics77 had been often used interchangingly l As we increase momentum of the particle7 eventually it exceeds the mass of the particle times the speed of light m0 and the particle becomes relativistic Therefore7 special relativity is an integral part of par ticle physics and we need to understand it thoroughly Even though special relativity has been discussed already in 110AB7 Id like to review it in the way that I can use it later on Relativity itself of course is one ofthe major revolutions in physics in 20th century7 together with quantum mechanics The way Einstein discovered it was based on theoretical consideration that Maxwell7s equation had unusual invariance7 Lorentz invariance He boldly stated that it must be the invari ance of our space and time It completely changed the way we view our space and time7 so intertwined that it is now called spacetime leading to exotic phenomena such as time dilation and Lorentz contraction lts marriage with quantum mechanics further led to a dramatic prediction of the existence of anti matter 1Of course7 particle physics experiments that do not use accelerators had also been quite successful as we will discuss in this course7 such as neutrino physics and cosmology7 and sometimes people distinguish these two terms 2 Gallilean space and time In the pre relativity view of the space and time7 different reference frames are related to each other by Gallilean transformations The notion of rela tivity77 itself was already there Namely7 laws of physics remain the same in any inertial reference frames that move relative to each other at constant speed2 Suppose the reference frame B is moving relative to the reference frame A by velocity Their origins coincide at t 0 If a particle moves at the position t in the frame A7 its position is simply given by impr m This coordinate change is called Gallilean transformation Just by taking the time derivative of both sides7 the velocities are related by 22 7V57V Namely7 velocities in two frames are related by a simple vector sum There is an implicit assumption in these equations that the time is com mon to both frames7 t t 3 It is normally not stated explicitly in discussions of mechanics7 and is taken for granted No matter which frame you are in7 time just ows the same7 right This naive statement of course underwent big revision under relativity 3 Michaelson Morley experiment Here7 we follow a bottom up approach to introduce relativity unlike the orig inal Einstein7s argument7 namely starting with an experimental evidence Michaelson and Morley7 using their sensitive interferometer mounted on hills7 demonstrated that the speed of light 0 is a fundamental constant inde pendent of the direction and the motion of the Earth itself3 It implies that 20f course accelerating frames exhibit different laws of physics that include the cen trifugal force and Coriolis force 3There were looking for the evidence of ether a postulated medium that transmits electromagenetic wavesi People in those days could not accept the fact that waves can change from one reference frame to another must be done in such a way that the speed of light remains the same This experimental fact goes against Eq 27 and hence the Gallilean trans formation itself What we have to do is to change the relationship coordinate transformation between two frames of reference We try to gure out what that is by requiring that the speed of light remains the same Consider small time and space intervals dt and d2 dx7dy7dz in the frame A What we would like to gure out is what the corresponding intervals are in the frame l37 dt and di For the light7 the time and space intervals are related by its speed7 d2 7 ltsz W ltsz 7 can lt4 The fact that the speed of light is common to both frames means this is also true in the frame l37 d2 dzgt2 dz2 dz adv lt5 ln Gallilean transformation Eq 137 the intervals in two reference frames are given by dt dt di 7 d2 7 m ICTJ If you insert these expressions into Eq 47 you nd cm2 7 d52 7 2i dfdt V2dt2 7 02 V2dt2 7 2i dfdt 8 Therefore Eq 5 is not satis ed propagate in vacuumlli All other waves people were familiar with had some kind of mediumi Sound through air7 ocean waves through water7 etci Ether77 was assumed to be weightless and frictionless as not to disturb the planetary motion over billions of years If there existed ether77 as the medium of the electromagnetic wave7 there must be its rest frame77 In any reference frame that moves relative to the rest frame7 the speed of light must be different By measuring the speed of light along with and orthogonal to the motion of the Earth7 their experiment was sensitive enough to demonstrate this point This expectation was wonderfully crushedi 4 Lorentz Transformation We are forced to look for coordinate transformations that preserve Eqs 45 The only assumption we make is that the transformation is linear Le it is a matrix multiplication on coordinates4 Because it is awkward to deal with all three spatial coordinates all the time let us focus on 2 direction only for the moment We will generalize the result to the case of three spatial coodinates later on We use cdt and dz so that both time and space coordinates have the same dimension The matrix multiplication is simply cdt 7 cdt 7 AttV AtzV cdt lt dz gt AW lt dz gt lt AV Aum dz 39 9 The Gallilean transformation Eq 7 corresponds to the matrix AV lt3 1 10 We start with requirements that do not concern with speed of light First obvious requirement is that an object at rest in frame A must appear to move with velocity 7V in frame B In this case dz 0 in frame A because it is not moving ln frame B dz dt 7V Therefore V 7 dz Aztcdt 7 Azt 11 c T cdt Attcdt 7 Ant The next requirement is less obvious If you go from frame A to B with relative velocity V and then combe back to A which is another coordi nate transformation with relative velocity 7V you havent done anything Therefore the matrices AV and A7V must be inverse of each other On the other hand the coordinate transformation A7V can be done by ip ping the z direction doing the transformation AV and then ipping the z direction back In other words A7V PAVP 12 where P ips the z coordinate 120 31gt lt13 4Nonlinear transformations appear in the general theory of relativity Einsteinls theory of gravity Such transformations are called general coordinate transformations 4 Therefore the requirement is that A7VAV PAVPAV I where I is the unit matrix By writing it out explicitly we nd i iAtzAzt AtzAttiAzz i 1 0 PAVPAV7ltAnltAttAu iAztAt Agz 7 0 1 14 The off diagonal elements must vanish and hence Att A Then the diag onal elements give the constraint Ag 7 AHA 1 Together with Eq 11 we can solve it up to a single unknown parameter Att 1 ii AV AttV lt ix VlAE gt 15 At this point it is easy to see that Gallilean transformation Eq 10 is a special case of this general form with Att 1 In fact if you require that time77 must be common to both frames you need Att 1 and Eq 10 is obtained as the unique choice under this requirement Because daily phenomena appear to support the idea that time77 is common no matter how fast you go people implicitly made this requirement and the Gallilean transformation followed as the only possibility Now comes the crucial requirement that the speed of light is the same in both frames The requirement is that cdt 27dz 2 0 if cdt27dz2 0 We subsitute dt and dz into Eq 5 and nd for light going right cdt dz 0 edit2 7 dz2 Attcdt Atzd22 7 Aztcdt Azzdz2 Att Atz2 7 Azt Azz2dz239 16 This combination must vanish Using the result obtained in Eq 15 we nd 0 A2 7 1 2 V 2 A2 177 ilt1iigt 0 17 t V Ag 6 lt gt A3 71 2 Aft 027 Therefore 18 and hence 1 5 At 19 We have now completely determined what coordinate transformation is con sistent with MichaelsoniMorley experiment As you have seen in the above discussion7 this is the unique choice This is the Lorentz transformation that allows us to take physical quantities from one reference frame to another It is conventional to use the notation V 6 i lt 1 20 c 7 1 gt 1 21 Y i f 7 62 Using this notation7 the Lorentz transformation is written as AV lt 7 35 gt 22 This is the expression we will use a lot in special relativity When we consider all three spatial coordinates7 the relative velocity be tween two frames is a vector V7 and we naturally de ne B Vc The de nition of y is similar7 y 1x1 752 Using the notation that B is a column vector7 the Lorentz transformation is given by v 775T gt a 23 7 17v AW lt The expression looks complicated7 but what is means is very simple The lower 3 gtlt 3 block has two terms They decompose any vector into a piece that is parallel to B by virtue of the projection BET32 and another that is orthogonal to B by its complement 756752 The rst term is not changed because the vector orthogonal to the relative velocity l7 is not affected7 while the second term is multipled by y 5 Implications of Lorentz Transformations There is one major difference between Gallilean transformation Eq 10 and Lorentz transformaion Eq 22 It is that the time is different in two frames after a Lorentz transformation Suppose you consider a time interval dt in the rest frame of an object dz 0 Then in a frame where the object is moving you nd dt ydt 24 This is the famous time dilation effect For instance a muon5 a charged particle similar to the electron decays with a lifetime 739 219703i000004 gtlt 10 6 sec At the velocity 1 06 it appears that it travels only over distance 60739 However because of the time dilation effect it survives over much longer time 7739 and hence the distance it can travel is 760739 This is why muons produced by the reaction of cosmic rays with atmosphere at an alti tude of 15720 km can reach the surface You may have seen a demo where a particle detector often a spark chamber literally keeps showing tracks of muons going downward Another related point is the Lorentz contraction In the rest frame of the atmosphere of thickness dz L the muon takes the time interval dt Lv Lc to traverse it In the rest frame of muon the Lorentz transformation gives cdtgtlt v iv gtltL gtltLv gt 25 dz 776 y L 0 39 The atmosphere of certain thickness comes rushing on top of him with veloc ity V and it takes the time interval Lv c to pass him Therefore he would conclude that the atmosphere had the thickness Lv c gtlt V LV It appears thinner than it is in its rest frame In the muon rest frame he can survive until the entire atmosphere passes him not because he lives longer but because the atmosphere is squashed In other words Lorentz contraction and time dilation are the same story told by different observers It is useful to check that Lorentz transformation reduces to Gallilean transformation for small velocities V ltlt 0 For daily phenomena typical times intervals dt are measured in seconds and typical distances dz in cen timeters Given 0 3 gtlt 1010cmsec 1 it follows that cdt gtgt dz Writing out Eq 22 we nd 1 V dt dt 7 idzgt 26 41 7 Vzc2 62 dz dzint 27 417V202 5We will discuss this particle a lot more later in this class In the rst line7 dz is completely negligible compared to dt7 while both dz and th can be comparable in the second line The prefactor y 1 1 7 VZc2 can be well approximated by 1 up to negligible corrections of 0V202 In the end the Lorentz transformation can be well approxi mated by dt dt 28 dz dz7 th 29 This is nothing but the Gallilean transformation Eq In other words7 Lorentz transformation indeed does reduce to Gallilean transformation at small velocities and Another important consequence of the Lorentz transformation is that the combination ch2 E cdt2 7 dz2 7 dy2 7 d22 30 is the same in any reference frames You can easily verify it using Eq 227 WT2 7 Edit2 7 ch2 7 dz2 7 dz2 7 vcdt 7 76612 7 M2 7 dz2 7 W 7 7666102 V2 7 YZBZXCd z 7 61902 7 dz2 7 YZ 7 7252d22 7 WV31 A quantity that does not change from one reference frame to another is called a Lorentz invariant or an invariant for short In this case7 d7 a proper time is an invariant lts physical meaning is the time interval in the rest frame of an object 6 Fourvector Notation We have seen that time and space get mixed up under Lorentz transfor mations It is therefore useful to consider time and space different compo nents of a single object7 a four component spacetime vector The notation is dz cdt7dz7dy7dz where the Greek index M runs from 0 dz0 cdt to 3 dz3 dz It is important that the index is a superscript This is called four vector notation Lorentz transformation is given by the matrix Eq 23 acting on the four vector dz written as a column vector Instead of using matrices and column vectors7 the following notation is also used often7 dz A dzquot 32 8 In this notation Af corresponds to the matrix A whose M V component is A A repeated index V in this case is always summed over from 0 to 3 This is often called Einstein7s convention776 Note that the sum is over a lower index and an upper index This is also an important aspect of the convention because of the reason we will see below Any four vector that transforms the same way as the spacetime four vector is said to be contravarian 77 As we will see in the next section energy and momentum of an object are combined into a contravariant four vector The proper time is an invariant It is useful to introduce a covarian 77 vector that has a lower index dx cdt idz idy idz 33 Using this notation the proper time can be written as ch2 dxpdx 34 The point is that if you sum over a lower and an upper index the indices cancel namely two indices combine to an invariant In the case of Lorentz transformation Eq 32 the lhs of the equation has only one index while the rhs three indices The point is that the lower index in AK and the upper index in dz are summed over and they cancel Effectively the rhs also has only one index leftover and both sides of the equation are allowed to be equated The way a contravariant and a covarint vector are related to each other is given by a metric tensor77 9W 1 0 0 0 0 71 0 0 WV 0 0 71 0 35 0 0 0 71 so that dx gwdxquot The inverse of the metric tensor has also the same form 1 0 0 0 V 0 71 0 0 9M7 0 0 71 0 7 36 0 0 0 71 6Somebody told me that it wasn t Einstein at all who invented this convention llm not a historian to refute or verify this claimi Any volunteer so that dx ngxV The inverse relation can be written as 9W9 6quot Kronecker7s delta which is an identity matrix Spacetime with the metric of this form is called Minkowsky space77 as opposed to Riemannian space77 whose metric is positive de nite 7 Energy and Momentum FourVector The action of a system must be Lorentz invaraint The only invariant for a point particle is given by the proper time d7 Then the natural candidate for the action of a point particle is S 7 m02d739 37 You will see the reason for the factor imcz shortly below Because of its de nition Eq 307 the action can be written explicitly as 2 1 d2 57 dtL d 27 2177 i dt me C m m0 CZ dt The Lagrangian is given by the integrand 1 i if 39 In the non relativistic limit7 we can Taylor expand the Lagrangian in to the second order and nd 1 L7m02lt177 1 2 202 1 i z gt imcz E77122 40 This is indeed the Lagrangian of a non relativistic point particle up to a constant term The overall coef cient imcz in the action was chosen to reproduce it Remember that the momentum is de ned by ELL 692 which gives mi in the non relativistc case In our relativistic case7 a 3 2 1g a piimc 177 m mc B 42 M 1 02 f 792262 7 10 17 41 It is useful again to check the non relativistic limit u ltlt 0 By Taylor expanding the expression for the momentum in 110 lflc the momentum is indeed 7m up to corrections suppressed by 17202 The energy is given by the Hamiltonian E 17 7 L In the non relativistic case it is 2 a 2 771 V 771 W A 1 1 E 77 7 43 p96 2 2 On the other hand in the relativistic case E 133 5 mczll 7 f2 m 22 WCZ Y 44 C V1 if 02 Again by taking the non relativistic limit you nd E mcz The rst term is the rest energy of a particle E m0 By comparing the expressions of the energy and the momentum it is easy to see that mlquot 2 7 132 771202 45 No matter how fast the particle is moving this combination is always the same In other words it is an invariant This observation suggests that we can de ne a contravariant vector E pf gypmpyypz 46 The above relationship can then be rewritten as pMp 771202 47 To see that this interpretation indeed makes sense let us look at its expression in the rest frame of the particle pr 7716707070 lt48 Then going to the frame moving with velocity 71 along the z direction the Lorentz transformation gives 1W mm 07 0777mm 49 11 This precisely agrees Ec and of the particle with what was obtained from the Lagrangian ln relativistic kinematics seen in particle physics the velocity is often very close to the speed of light and it is not very useful to talk about it Moreover the velocity itself is not a conserved quantity under collision processes while the energy and momentum are Therefore we almost exclusively talk about energy and momentum very rarely about the velocity Especially when we discuss massless particles photon or particles with tiny masses electron neutrinos the velocity is always approximately the speed of light yet energy can be any value The formulae useful to remember are 1 E M02172 771204 7V E2 7 771204 50 c If you need the velocity you can get it as 2 l 7 51 1 c E It is amusing that the plane wave solution to the Schrodinger equation ei Ehe iEth can be written as e iWWh Secretly the Schrodinger equation gave us a Lorentz invariant expression for the wave function7 8 Doppler Shift As an application of Lorentz transformation and four vectors let us discuss Doppler shift of light emitted from moving bodies for example far away stars that are moving away from us because of expansion of Universe8 The plane wave solution to Maxwell equation has time and space depen dence e m lk39w The exponent suggests that we can de ne contravariant wave four vector kf LUC The plane wave solution then has manifestly Lorentz invariant form e ik w p This is enough information for us to gure out the Doppler shift of light 7Of course the nonrelativistic Schrodinger equation does not give the relativistic re lation between the energy and the momentumi 8T0 correctly calculate the redshift in expanding Universe however general relativistic effects must also be taken into account 9If you compare it to the quantum mechanical plane wave solution in the previous section the relationship is obvious k pHhi 12 The dispersion relation of light is w For light propagating along the z direction therefore the wave four vector is simply kf LU612 wc100 1 Suppose w is the frequency of light in the rest frame of the emitter If the light emitter is moving away from us with velocity 1 all we need to do is to Lorentz transform this four vector Using by now familiar matrix Eq 22 we nd k wc y 7 76 0 07 7 76 In other words the new angular frequency is related to the one in the rest frame of the light emitter by w wwew w wi 11 lt52 This must be a familiar formula from Physics 70 Because light emitted from atoms has de nite spectrum that can be mea sured in the laboratory and spectroscopy can be done very accurately mea suring redshift of distant stars is a very powerful tool in astronomy 9 Natural Unit In the world of particle and nuclear physics where both relativity and quan tum mechanics play crucial roles it is cumbersome to keep writing the speed of light 0 and the reduced Planck constant h It is customary to set c h 1 Why is this allowed Remember in the cgs system also true in the MKS sytem the funda mental units have dimensions of length L mass M and time Other units are derived from these three For example ampere is de ned by the force between two currents and it does not have to be a new unit Indeed in esu unit in the cgs system electric charge and current are de ned in terms of cm g and s only Given the three fundamental dimensions we have the freedom of relating them among each other For example the length can be measured in the unit of time using the speed of light 0 which has dimension LT l ln daily life it is extremely inconvenient to talk about length of 33ps77 if you mean just a centimeter77 Note ps is pico second 10 12 sec But in the world of particle physics it is not inconvenient at all In fact a typical size of a particle detector 1710m translates to nanoseconds which immediately put challenging demands on electronics This way L and T are now equivalent 13 Similarly7 the Planck constant has dimension LZMT l It can be used to relate the dimension of mass to other two Most physicists use electronvolt eV as the unit for energy Using the constant he 197Merm MeV is 106eV7 fm is femto meter 10 15m7 we can relate the length and the energy Remember this constant Any particle physicist remembers this number For example7 we often refer to the mass of electron as me 0511 MeVCZ For short7 we dont quote 02 We just say the mass as 0511 MeV As long as it is understood that we always use 0 and h to relate different dimensions7 there is no source of confusion When the value is quoted in MeV7 in order to get the dimension of mass7 all you need to gure out is that you need a factor of 102 remember E mczl If you want7 you can then work out the mass in SI unit7 by using conversion constans 6V 160gtlt10 19 J and c 300gtlt108 ms as me 0511 gtlt106eV160gtlt10 19JeV300gtlt108ms2 908gtlt10 31kg Actually7 it is 911 gtlt 10 31kg We made this error because we kept only three signi cant digits If the electron is moving with energy 1 MeV7 we work out the momentum by p E2 7 m2 0860MeV 53 What we mean is that the momentum is 0860 MeVc7 but we dont write 0 Only when you want to convert the momentum to the SI unit7 you need to remember that there must be a factor of 10 Otherwise7 you can consistently drop 0 everywhere This way7 we quote values for energy7 momentum7 mass all in eV unit If you want the wave length A7 we of course use de Broglie7s formula A hp 27Thp But if you use natural unit7 we quote p in eV7 which is the same thing as writing it as A 27Thccp Then the only constant you need to remember is he We immediately obtain A 1440 fm This is the resolution of an experiment where 1 MeV electron is scattered There is another fundamental constant which can be used to eliminate any units Newton7s constant GN It has the dimension LSM lT Z It can be used to construct a unit for energy7 Eglamk hogGA 122 gtlt1019GeV2 called Planck energy Correspondingly7 the Planck length is 113an hcEplamk 161 gtlt10 33 cm7 an extremely short distance Then any physical quantity can be expressed in terms of pure numbers with no dimensions However7 this energy is so enormous that it is not used as a unit even in particle physics On the other hand7 this energy length scale signals where quantum effects of gravity need to be considered In fact7 the candidate theory of quantum 14 gravity7 string theory7 is often discussed using this unit10 Note that it is not only particle and nuclear physicists who choose the unit system for convenience Atomic physicists also use the same freedom to x all three dirnensions7 by setting 6 m5 h 1 It is called atornic unit 10Actually7 What is used as the unit in string theory is not precisely the Planck unit7 but a combination of string coupling constant and Eplaanx 87r is set to unity 15

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