Advanced Quantum Mechancs Rajdeep Sensarma! sensarma@theory.tfr.res.n ecture #9 QM of Relatvstc Partcles
Recap of ast Class Scalar Felds and orentz nvarant actons Complex Scalar Feld and Charge conjugaton Spn 1/ representatons and Weyl Spnors Transformaton propertes of Weyl spnors
Rules for Possble Actons The dynamcs of relatvstc partcles s obtaned from the acton of the system, whch s a functon of the felds (operator valued functons of space-tme) and ther space-tme dervatves.! In partcular transton ampltudes can be wrtten as a functonal ntegral over feld confg. wth each confg contrbutng e S Choose felds transformng accordng to partcular rreps of the orentz group. The transformaton propertes of the felds are frame ndependent, e.g. a scalar feld remans a scalar feld n all frames. Construct possble orentz nvarant quanttes (orentz scalars) out of felds and ther dervatves. These provde possble agrangan Denstes for descrbng the system. For free partcle, the acton should be quadratc and Poncare nvarant. It should nvolve atmost nd order dervatves. Our Am: Construct possble orentz scalars out of Weyl Spnors
Weyl Felds and Grassmann Numbers Snce the (H) Weyl Feld transforms accordng to (1/,0) rrep, we should be able to construct a orentz Scalar by takng antsymmetrc products of H Weyl Felds SU() decomposton rule 1/ X 1/ = 0 + 1 For ths we would lke to consder two Weyl spnors (x, t) and (x, t) and consder ther products. We would lke to construct products whch are orentz nvarant. Under T: T! T U T ( ) U ( ) = T Snce U T ( ) U ( ) = Snce the scalar s obtaned by antsymmetrzng the product (look up exchange symmetry and SU()), we should get 0 f χ s same as 0 T =( 1, ) 0 1 = 1 + 1 = 0 f s a complex no. However, f the feld s a Grassmann number (ant-commutng number), there can be a non-trval scalar representaton formed out of the spn 1/ feld.
Grassmann Numbers A bnary representaton of an nteger Representng numbers: Start wth basc objects (generators) n for all non-negatve nteger n Defne combnaton rules: a) m n = m+n b) The objects commute wth each other m n = n m Arbtrary lnear combnatons of these generators wth co-effcents (0,1) represent ntegers Complex Numbers: Start wth basc objects (generators) 1 and Defne combnaton rules: 1 * = * 1 = * = -1 1* 1 =1 A complex no. can be represented as arbtrary lnear combnaton of { 1,} wth real co-effcents Grassmann Numbers: Start wth set of n ant-commutng generators + =0 =0 A Grassmann number s an arbtrary lnear comb. of { 1, α1,. αn, α1 α., α1 α αn } wth complex co-eff. E.g. wth generators 1 and = a 0 + a 1 1 + a + a 1 1
Grassmann Numbers Addng Grassmann Numbers: Thnk of { 1, α1,. αn, α1 α., α1 α αn } as ndependent unt vectors and add component-wse coeffcents Multplyng Grassmann Numbers: Multply each component wth every other component, and keep track of + =0 =0 Complex conjugaton of Grassmann Numbers: Select a set of n generators and assocate a conjugate generator to each ( ) = ( ) = ( ) = = The generators commute wth complex numbers = Defne conjugaton as an operaton whch conjugates both generators and co-effcents.
Grassmann Numbers Functons of Grassmann numbers Stck to generators and * Any analytc fn. of * f( )=f 0 + f 1 Any analytc fn. of g( ) =g 0 + g 1 Any analytc fn. of and * A(, )=a 0 + a 1 +ā 1 + a 1 Grassmann dervatves: Identcal to complex derv., BUT, for the dervatve to act on the varable, the varable has to be antcommuted tll t s adjacent to the dervatve operator. E.g.: Grassmann Integrals: Defne @ ( )= @ ( )= d 1 = d 1=0 @ @ A(, )= a 1 = @ d = @ A(, ) Grassmann dervatves antcommute d =1 ke dervatves, the varable has to be antcommuted tll t sts adjacent to ntegral operator E.g.: d ( )=f 1 d d A(, )= a 1 = d d A(, ) Scalar product of Grassmann Fn.s : g( ) =g 0 + g 1 h( ) =h 0 + h 1 hh g = d d e h ( )g( ) = d d (1 )(h 0 + h 1 )(g = h 0g 0 + h 0 + g 1 ) 1g 1
Weyl Felds and orentz Scalars If Weyl felds are represented by Grassmann numbers, we have seen that T and R T R are quadratc orentz scalars that can be formed. These would ndcate the possblty of wrtng down mass terms for Weyl Fermons. However, Consder the transformaton propertes of where s a left-handed Weyl feld! U ( ) = U ( ) = U R ( ) U ( ) = U R( ) Thus * transforms lke a rght handed Weyl Feld Takng h.c. of above equaton T! T U R ( ) T! R T R! Thus the possble mass term has the form m R or m R The mass term mxes left-handed and rght handed Weyl felds. et us frst descrbe massless Fermons wth Weyl felds and we wll come back to massve felds later.
orentz vectors from Weyl Felds It s possble to construct four-vectors (transf. acc. to (1/,1/) rrep) from products of Weyl Felds. If we can construct orentz vectors, we can take ts norm, contract wth other orentz vectors lke etc. to form possble orentz scalars Consder the transformaton propertes of Under rotatons, represented by Untary operators, ths s a scalar. Under Boosts, U ( ) = e ~ (~! ~ )! e 1! e 1 ~ ~ ~ ~! e ~ ~ ' ~ ~ = For Infntesmal Boosts! e 1 ~ ~ e 1 ~ ~ ' 1 j ( j + j ) = {, j } = j
orentz vectors from Weyl Felds For Infntesmal Boosts! e ~ ~ ' ~ ~ =! e 1 ~ ~ e 1 ~ ~ ' 1 j ( j + j ) = Consder µ wth 0 beng X dentty matrx Under Boosts, ths behaves lke a 4-vector µ = µ wth 0 = Further, behaves as a 3-vector under rotatons So n all µ = (, ) s a orentz 4-vector Smlarly R µ R = ( R R, R R) s a orentz 4-vector Snce ( R R) = R R the above Grassmann blnears are real (n the sense of conjugaton of G No.s)
orentz Scalars and Weyl Acton The smplest orentz scalar s formed by contractng the V wth @ µ ( ) µ or µ @ µ ( ) The secret behnd Drac s magc of 1st order orentz nvarant eqn. Real near combnaton 1 h µ @ µ ( ) @ µ ( ) µ 1 µ~ @µ Quadratc Acton for eft Handed Weyl Spnors S = d 4 x 1 h µ @ µ ( ) @ µ ( ) µ d 4 x 1 µ~ @µ Smlarly for Rght Handed Weyl spnors @ µ ( ) µ R R or R µ@ µ ( R ) or 1 h R µ@ µ ( R ) @ µ ( ) µ R R 1 R µ~ @µ R Quadratc Acton for Rght Handed Weyl Spnors S = d 4 x 1 h R µ@ µ ( R ) @ µ ( ) µ R R d 4 x 1 R µ~ @µ R Gradent Terms > Poncare Invarant
Feld Equatons for Weyl Felds Quadratc Acton for eft Handed Weyl Spnors S = d 4 x 1 µ~ @µ = d 4 x µ @ µ upto boundary terms Feld Equaton: Multplyng by, µ @ µ =0 ( 0 @ t + @ ) =0 [@ t ~ ~p ] =0 Smlarly for RH Weyl Fermons [@ t + ~ ~p ] R =0 Snce the Fermons are massless t > E = p [1 ~ ~p p ] =0 [1 + ~ ~p p ] R =0 The Soluton of Weyl Equaton are egenstates of the helcty operator gven by where s s the spn operator, wth egenvalues ± 1 ~s ~p p Paul ubansk vector W 0 = ~s ~p related to helcty n ths case
Party and Weyl Fermons The helcty operator s a pseudoscalar under O(3),.e. t changes sgn under a party transformaton. So a H Weyl spnor wth helcty +1 wll transform to a RH Weyl spnor wth helcty -1 under party transformaton. A party nvarant theory (even for massless Fermons) requres both H and RH Weyl felds.
Massless Drac Fermons [ (1/,0) + (0,1/) We have seen that party mxes the left and rght handed Weyl Fermons. et us keep both felds and construct a 4 component spnor. We want to wrte a party nvarant theory Ths transforms as (1/,0) (0,1/) = R Ths s called a Drac spnor n the chral bass How does party act on the Drac spnor? P = P R = R = 0 0 = 0 1 1 0 1 s X dentty matrx 0 =1 1 s 4 X 4 dentty matrx et us now wrte the theory of a left handed and a rght handed Weyl Fermon together S = d 4 x 1 [ µ~ @µ + R µ~ @µ R ] = d 4 x 1 [ 0 µ @µ ~ ] = d 4 x 1 [ µ~ @µ ] where = 0 0 = 0 { µ, } =g µ Clfford Algebra Paul conjugate
Massless Drac Fermons [ (1/,0) + (0,1/) Check that we have a orentz nvarant acton Check that µ Transforms lke 4 vector To do ths, we need to fnd the form of orentz generators J and K and R spnors transform as spn 1/ under rotn. J = 0 0! K = 0 0! and R spnors transform dfferently under boosts 0J = J 0 0 K = K 0 K nvolves one space and 1 tme co-ord > J s pseudovector under rotn. changes sgn under party.! U( )! U ( ) and 0! U ( ) 0 wth U( ) = e ( J ~! ~ + K ~ ~ ) U ( ) = e ( J ~! ~ K ~ ~ ) U 1 ( ) = e ( ~ J ~! + ~ K ~ ) Note: γ 0 changes form under T, but mantans ts antcommutaton wth K n new bass Snce γ 0 changes sgn of K whle commutng across, but keeps J unchanged, commutng t across U wll convert t to U -1 0! U ( ) 0 = 0U 1 ( ) Thus s a orentz scalar
Massless Drac Fermons [ (1/,0) + (0,1/) Consder the matrx µ = 4 [ µ, ]= 4 ( µ { µ, }) = ( µ g µ 1 4 ) [ µ, ]= [ µ, ] = ( µ [, ]+[ µ, ] ) = ( µ [, ] [, µ ] ) = ( µ µ g µ +g µ ) = ([ µ, ] µ g + g µ ) So [ µ, ]= [ µ, ]=(g µ g µ ) [ µ, ]= [ µ, ] = ([ µ, ] + [ µ, ]) = (g µ g µ + g µ g µ ) = (g µ g µ + g µ g µ ) Γ μν satsfes the e Algebra for orentz generators So S µ = 4 [ µ, ] These relatons only use Drac Algebra and not specfc forms for γ matrces
Massless Drac Fermons [ (1/,0) + (0,1/) S µ = 4 [ µ, ] [S µ, ]= [ µ and, ]=(g µ g µ ) γ transforms as a orentz 4-vector µ transforms as a orentz 4-vector S = d 4 x 1 [ µ~ @µ ] s a orentz scalar These relatons only use Clfford Algebra and not specfc forms for γ matrces 4 x 4 matrces satsfyng Clfford Algebra s not unque In fact any untary transformaton wll keep { µ, } =g µ nvarant. Thus there are many equvalent bass to wrte Drac spnors. The form of the spnors as well as the γ matrces depend on the bass, but the form of the acton s nvarant. Chral bass = R Charge conjugated spnor c = R Majorana spnor M = Real spnor (Real G No.s) γ matrces are purely magnary
Massve Drac Felds et us now come to the queston of a mass term Snce s a orentz scalar m = m( R + R ) s a possble real orentz scalar mass term Ths s not the only possble quadratc orentz scalar We had earler shown that R and R are ndvdually orentz scalars An ndependent scalar can be formed out of the dfference of ths two terms m( R R ) Defne projecton operators to obtan Weyl Fermons from Drac Fermons In the chral bass, followng projecton operators project the Drac spnor nto eft(rght) handed Weyl Fermons (R) = 1 (1 ± 5) 5 = 1 0 0 1 5 = 0 1 3 = 4 µ µ γ 5 transforms as a orentz scalar, but changes sgn under party.e. transforms as a pseudo-scalar. Note: Explct for of γ 5 wll be dfferent n dff. bass, but ts relaton wth γ holds n all bass
Massve Drac Fermons m( R R )! m 5 So the most general Drac acton for a massve feld has the form S = 1 d 4 x[ µ~ @µ + m m 0 5 ] If party s a good symmetry of the system m = 0 S = 1 d 4 x [ µ~ @µ + m] = 1 d 4 x [ µ @ µ + m] upto surface terms The Saddle pont equaton for ths acton s [ µ @ µ m] =0 Multply by γ 0, and use (γ 0 ) =1 [( 0 ) @ t + 0 ~ r m 0 ] =0 0 ~ = ~, 0 = @ t =[ ~ r+ m ] Drac Equaton of 1 patcle Rel. QM
Global Symmetres of Drac Acton S = 1 d 4 x [ µ~ @µ + m] = 1 d 4 x [ µ @ µ + m] Global phase rotaton! e Chral Transformaton! e 5 The acton s nvarant under these transformatons The conserved Noether current s gven by j µ = µ j µ 5 = µ 5 and the correspondng conserved charges are Q = d 3 x 0 = d 3 x [ + R R] Q 5 = d 3 x 0 5 = d 3 x [ R R]
Thngs we have not touched Constructng creaton/annhlaton operators and Fock space from felds Calculatng Expermentally measureable quanttes Scatterng ampltude as transton ampltude > Calculaton through fn ntegrals. Fnte Chemcal potental > fnte densty of partcles Imagnary tme and fnte temperature calculatons > Correlaton fn. and response fn. Interactng theores Symmetry Consderatons and possble nteracton terms. Perturbaton Theory calculaton of n pont functons > scatterng, correlaton fn. etc. Saddle Ponts, symmetry breakng and effectve theores Renormalzaton the other gudng prncple Wat for QFT-I next semester.