Transcript Slide 1
Utrecht University Some remarks on an old problem Gerard ‘t Hooft The Many Faces of QFT, Leiden 2007 P. van Baal, Twisted Boundary Conditions: A Non-perturbative Probe for Pure Non-Abelian Gauge Theories thesis: 4 July, 1984. SPIRES: Lattice regularization of gauge theories without loss of chiral symmetry. Gerard 't Hooft (Utrecht U.) . THU-94-18, Nov 1994. 11pp. Published in Phys.Lett.B349:491-498,1995. e-Print: hep-th/9411228 The Many Faces of QFT, Leiden 2007 Gauge theory on the lattice: U(12) Site x=1 Plaquette 1234 4 Pe 1 2 1 2 Link 23 3 The Many Faces of QFT, Leiden 2007 ig A dx U(12) Pe 1 ig Tr (U (12)U (23)U (34)U (41) ) Tr P (e 1 2 2 4 ig C Tr (iga F12 g a F 2 2 2 12 A dx A dx 1234 ) After symmetrization : Tr (F F ) 1 4 1 2 Tr (UUUU ) The Many Faces of QFT, Leiden 2007 ) The Fermionic Action (first without gauge fields) : U12 (x ) 12 11 Dirac Action ( x2 ) ( x1 ) L ( x1 ) ( ) m ( x1 ) a links However, in the limit m 0 , the equation has ( ( x e ) ( x)) 0 st several solutions besides the vacuum solution C '( x1, , x4 ) C1 (1) x ( x1, , x4 ); C1 51 1 since C1 1 1C1 (and same for 2, 3 ) The Many Faces of QFT, Leiden 2007 : Wilson Action ( x e ) ( x e ) L ( x1 ) m ( x1 ) (1 ) (1 ) a a links This forces us to treat the two eigenvalues of separately, and species doubling is then found to disappear. Effectively, one has added a “mass renormalization term” However, now chiral symmetry has been lost ! Nielsen-Ninomiya theorem The Many Faces of QFT, Leiden 2007 It could not have been otherwise: even in the continuum limit 5 invariance is broken by the Adler-Bell-Jackiw anomaly. However, in the chiral limit, M 0 , the symmetry pattern is SU ( N F ) L U (1) L SU ( N F ) R U (1) R ABJ anomaly SU ( N F ) L SU ( N F ) R U (1)V Lattice SU ( N F )V U (1)V Can one modify lattice theory in such a way that SU ( NF )L SU ( NF )R U (1)V symmetry is kept? The Many Faces of QFT, Leiden 2007 The BPST instanton (A.A. Belavin, A.M. Polyakov, A.S. Schwartz and Y.S. Tyupkin) SU (2) 3 The Many Faces of QFT, Leiden 2007 The massless fermions LEFT time Fermi level RIGHT Instanton The Many Faces of QFT, Leiden 2007 The fermionic zero-mode: In Euclidean time: In Minkowski time: e e eit eit negative energy positive energy Thus, the number zero modes determines how many fermions are lifted from the Dirac sea into real space. Left – right: a left-handed fermion transmutes into a right-handed one, breaking chirality conservation / chiral symmetry The Many Faces of QFT, Leiden 2007 The instanton breaks chiral symmetry explicitly: m2 m2 implies U (2) U (2) SU (2) SU (2) U (1) m2' mK2 implies U (3) U (3) SU (3) SU (3) U (1) The Many Faces of QFT, Leiden 2007 Each quark species makes one left - right transition at the instanton. uLeft uRight dRight dLeft sLeft sRight cLeft cRight mcharm The Many Faces of QFT, Leiden 2007 The interior is mapped 4 onto SU (2) 3 The number of left-minus-right zero modes of the fermions = the number of instantons there. Atiyah-Singer index The Many Faces of QFT, Leiden 2007 How many “small” instantons or anti-instantons are there inside any 4-simplex between the lattice sites? These numbers are ill-defined ! The Many Faces of QFT, Leiden 2007 The number of instantons is ill-defined on the lattice! Therefore, the number of fermionic modes cannot depend smoothly on the gauge-field variables U ( x) on the links! If one does keep this number fixed, one will never avoid the species-doubling problem. Domain-wall fermions are an example of a solution to the problem: there is an extra dimension, allowing for an unspecified number of fermions in the Kaluza-Klein tower! Is there a more direct way ? The Many Faces of QFT, Leiden 2007 We must specify # ( instantons) inside every 4-simplex. This can be done easily ! Construct the gauge vector potential from the lattice link variables A ( x) at all x , starting U ( x) (defined only on the links) def Step #1: on the 1-simplices U ( x) e iagA Note: this merely fixes a gauge choice in between neighboring lattice sites, and does not yet have any physical meaning. Next: Step #2: on the 2-simplices The Many Faces of QFT, Leiden 2007 U I First choose local gauge : A2 ( x1 , x2 ) A ( x1 / a) U I 2 U e iagA 1 F12 a A U I Here, we may now choose the minimal flux F , which means that all eigenvalues must obey: A ag This is unambiguous only in the elementary, faithful representation, which means that we have to exclude invariant U(1) subgroups – the space of U variables must be simply connected – we should not allow for a clash of the fluxes ! Then subsequently, if so desired, gauge-transform back This procedure is local, as well as gauge- and rotation-invariant ( The subset of gaugetransformations needed to rotate is Abelian ) The Many Faces of QFT, Leiden 2007 Step #3: on the 3-simplices We have A ( x) on the entire boundary. Extend the field in the 3-d bulk by choosing it to obey sourceless 3-d field eqn’s (extremize the 3-d action 3 d x ( Fij ( x) Fij ( x)) , and in Euclidean space, take its absolute minimum ! ) This prescription is gauge-invariant and it is local !! Step #4: on the 4-simplices Exactly as in step #3, but then for the 4-simplices. Taking the absolute minimum of the action here fixes the instanton winding number ! The Many Faces of QFT, Leiden 2007 Thus, there is a unique, gauge-independent and local way to define A ( x) as a smooth function of x starting from the link variables U ( x) In principle, we can now leave the fermionic part of the action continuous: L ( m ( igA( x)) ) fermion Our theory then is a mix of a discrete lattice sum (describing gauge fields and scalars) and a continuous fermionic functional integral. The fermionic integral needs no discretization because it is merely a determinant (corresponding to a single-loop diagram that can be computed very precisely) The Many Faces of QFT, Leiden 2007 log det ( D0 ig A) log det D0 i log i 1 C1 A 1+ + + + + ··· The first four diagrams can be regularized in the standard way – giving only the standard U(1) anomaly The sum over the higher order diagrams can be bounded rigorously in terms of bounds on the A fields. (Ball and Osborn, 1985, and others) - one might choose to put the fermions on a very dense lattice: a fermion a gauge , to do practical lattice calculations, but this is not necessary for the theory to be finite ! The Many Faces of QFT, Leiden 2007 The procedure proposed here is claimed to be non local in the literature. This is not true. The extended gauge field inside a d -dimensional simplex is uniquely determined by its (d – 1) -dimensional boundary The Many Faces of QFT, Leiden 2007 The prescription is: solve the classical equations, and of all solutions, take the one that minimizes the total action. However, imagine squeezing an instanton in a 4-simplex, using a continuous process (such as gradually reducing its size). As soon as a major fraction of the instanton fits inside the 4-simplex, a solution with different winding number will show up, whose action is smaller. The Many Faces of QFT, Leiden 2007 The most essential part of the gauge field extrapolation procedure consists of determining the flux quanta on the 2-simplices, and the instanton winding numbers of the 4-simplices. We demand them to be minimal, which usually means that the Atiyah-Singer index on one simplex 2 g 4 F F d x 2 32 1 2 → the gauge field extrapolation procedure itself is discontinuous ! Depending on the configuration of the link variables U, the number of instantons within given 4-simplices may vary discontinuously. This is as it should be! The Many Faces of QFT, Leiden 2007 Lattice regularization of gauge theories without loss of chiral symmetry. Gerard 't Hooft (Utrecht U.) . THU-94-18, Nov 1994. 11pp. Published in Phys.Lett.B349:491-498,1995. e-Print: hep-th/9411228 We claim that this procedure is important for resolving conceptual difficulties in lattice theories. The END The Many Faces of QFT, Leiden 2007