Introduction Non-Abelian interactions Conclusion Mixed symmetry tensors in the worldline formalism James Edwards NBMPS 44 - York September 2015 Based on arXiv:1411.6540 [hep-th] and arXiv:1510.xxxx [hep-th] Mixed symmetry tensors in the worldline formalism
Introduction Non-Abelian interactions Conclusion Outline Introduction 1 Motivation Worldline formalism Non-Abelian interactions 2 Gauge Holonomy Results Generalisation Conclusion 3 Mixed symmetry tensors in the worldline formalism
Introduction Motivation Non-Abelian interactions Worldline formalism Conclusion Introduction Calculations in quantum field theory must often be done in a perturbative expansion. Not always desirable: The number of Feynman diagrams increases rapidly with the order of the coupling constant. Gauge invariance not manifest. Strong coupling regime largely inaccessible. Non-trivial matter multiplets lead to complicated Feynman rules The so-called worldline formalism of quantum field theory offers significant computational advantages over conventional perturbative approaches. Fewer diagrams at any given order. Feynman parameterised expressions and transverse photon appear earlier and more naturally. First quantised – it’s just quantum mechanics! Mixed symmetry tensors in the worldline formalism
Introduction Motivation Non-Abelian interactions Worldline formalism Conclusion Successes Worldline approaches have been used to address many problems N-point scattering amplitudes at one loop order and beyond Euler-Heisenberg action for a constant electromagnetic field Gravity-matter coupling and calculation of the gravitational effective action Graviton / photon production Trace anomalies Non-commutative quantum field theory Higher spin fields The value of this first quantised approach is only just starting to be recognised and we are seeing a huge resurgence in interest in these techniques. Mixed symmetry tensors in the worldline formalism
Introduction Motivation Non-Abelian interactions Worldline formalism Conclusion Worldline formalism The worldline formalism of quantum field theory relates the field theory to a set of one dimensional curves interpreted as the worldlines of particles described by ordinary quantum theory. Strassler [ 1 ] reformulated scalar and spinor QFT and derived the Bern-Kosower “Master Formula” without recourse to string theory. Integrating over matter fields gives effective action: �� � �� � � ¯ � d 4 x ¯ Γ [ A ] = log D ΨΨ exp − Ψ ( γ · D − m ) Ψ = − 1 � ( γ · D ) 2 + m 2 � 2Tr log 1 Nucl. Phys. B385 Mixed symmetry tensors in the worldline formalism
Introduction Motivation Non-Abelian interactions Worldline formalism Conclusion Worldline formalism The functional trace can be written as the transition amplitude for a quantum particle to traverse a closed path in some proper time T : we integrate over all such paths and all proper times. This leads to the effective transition amplitude (henceforth take m = 0 for simplicity) � ∞ � dT Γ [ A ] ∝ D ( w, ψ ) exp ( − S point ( w, ψ )) W [ A ] T 0 where � 1 ω 2 S point = 1 ˙ T + ˙ ψ · ψ dτ and 2 0 � 1 � � �� � dω · A A T A + iqT dτ ψ µ F A µν T A ψ ν W [ A ] = tr P exp iq 2 0 Mixed symmetry tensors in the worldline formalism
Introduction Gauge Holonomy Non-Abelian interactions Results Conclusion Generalisation The Wilson loop � 1 � � �� � dω · A A T A + iq dτ ψ µ F A µν T A ψ ν W [ A ] = tr P exp iq 2 0 How can we include gauge interactions and path ordering in the worldline theory? Introduce “colour” fields ˜ φ r and φ r with Poisson brackets { ˜ φ r , φ s } PB = − iδ rs . Define R A ≡ ˜ φ r T A rs φ s and note that { R A , R B } PB = if ABC R C . We specify the dynamics of these new fields with the action � 1 dτ ˜ φ ( d τ + A ) φ. 0 The Green function of these fields is G ( τ, τ ′ ) ∼ Θ ( τ − τ ′ ) which is just what is needed to generate the path ordering along the worldlines. Mixed symmetry tensors in the worldline formalism
Introduction Gauge Holonomy Non-Abelian interactions Results Conclusion Generalisation The Fock space We promote ˜ φ and φ to creation and annihilation operators and Poisson brackets become commutators or anti-commutators. The Hilbert space is described by wavefunction components which transform in fully (anti)-symmetric products of the representation of the T R : Ψ( x, ˜ φ ) = Ψ( x ) + Ψ r 1 ( x )˜ φ r 1 + Ψ [ r 1 r 2 ] ( x )˜ φ r 1 ˜ φ r 2 + · · · +Ψ [ r 1 r 2 ..r N ] ˜ φ r 1 ˜ ... ˜ φ r 2 φ r N φ r 1 + Φ ( r 1 r 2 ) ( x )˜ φ r 1 ˜ φ r 2 + · · · +Φ ( r 1 r 2 ..r p ) ˜ φ r 1 ˜ φ r p + ... Φ( x, ˜ φ ) = Φ( x ) + Φ r 1 ( x )˜ ... ˜ φ r 2 Project onto a given representation by gauging a U (1) symmetry which constrains the occupation number of the colour fields [ 2 ] . These new degrees of freedom generate all interactions. � 1 � � � � � − 1 ˜ dτ ˜ Z [ A , θ ] = D φ, φ exp φ ( d τ + A + θ )) φ 2 0 2 arXiv:0503.155[hep-th] Mixed symmetry tensors in the worldline formalism
Introduction Gauge Holonomy Non-Abelian interactions Results Conclusion Generalisation Wilson-loops If the colour fields are taken to be Grassmann valued, we find ) e 2 iθ + tr W ( ) e 3 iθ + . . . + Z N [ A , θ ] ∝ tr W ( · ) + tr W ( . ) e ( N − 1) iθ + tr W ( · ) e Niθ . tr W ( . If the colour fields are instead bosonic , we have ) e iθ + tr W ( ) e 2 iθ + tr W ( ) e 3 iθ + . . . Z N [ A , θ ] ∝ tr W ( · ) + tr W ( ) e piθ + . . . . + tr W ( ·· � 2 π 2 π e − iθn picks out the representation with exactly dθ Integrating against 0 n fully (anti-)symmetrised indices. Mixed symmetry tensors in the worldline formalism
Introduction Gauge Holonomy Non-Abelian interactions Results Conclusion Generalisation Arbitrary representations Rather than being restricted to fully (anti-)symmetric representations we would like to project onto an arbitrarily chosen irreducible representation. We can achieve this by introducing multiple families of the colour fields: � 2 π ω 2 φ, φ ] = 1 dτ ˙ � ˜ � S [ ω, ψ, ˜ T + ψ · ˙ ψ + � F r ˙ r + ˜ φ k φ k φ k r A rs φ k . k =1 s 2 0 These F fields span a Hilbert space described by wavefunctions transforming in the tensor product of the representations associated to each family � . . . Ψ( x, ˜ φ ) ∼ . ⊗ . . . ⊗ . ⊗ . { n 1 ,n 2 ,...n F } ���� ���� ���� n F n 2 n 1 � Φ( x, ˜ φ ) ∼ ·· ⊗ . . . ⊗ ·· ⊗ ·· . � �� � � �� � � �� � { n 1 ,n 2 ,...n F } n F n 2 n 1 Mixed symmetry tensors in the worldline formalism
Introduction Gauge Holonomy Non-Abelian interactions Results Conclusion Generalisation Irreducibility There is now a richer U ( F ) symmetry rotating between the families of colour fields which can be used to construct a projection onto a single irreducible representation. We need worldline gauge fields a jk ( τ ) for the generators of this symmetry group L jk = ˜ φ r j φ kr but only for k � j This partial gauging allows for the introduction of independent Chern-Simons terms fixing the occupation number of each family � 1 � S = dτ a kk ( τ ) n k 0 k The off-diagonal generators impose further constraints on the physical states, selecting the representation with desired symmetry from the tensor product decompositions on the previous slide. Mixed symmetry tensors in the worldline formalism
Introduction Gauge Holonomy Non-Abelian interactions Results Conclusion Generalisation Worldline theory One can gauge fix by setting a jk = diag( θ 1 , θ 2 , . . . θ F ) where the θ k are moduli to be integrated over. The Faddeev-Popov determinant gives a measure for these moduli � 1 − e − iθ j e iθ k � µ ( { θ k } ) = � j<k µ ( { θ k , θ j } ) = � j<k We use path integral quantisation on this gauge slice: � ∞ � 2 π F � dT dθ k � 2 π � ω 2 2 π e − in k θ k µ ( { θ k } ) Z ( F ) [ A , { θ k } ] T + ψ · ˙ D ω D ψ e − 1 ˙ ψ 2 0 T 0 0 k =1 The partition function of the extended colour fields is F � � � 2 π Z ( F ) [ A , { θ k } ] = � ˜ � dτ ˜ φ k ( ∂ e − 1 ∂τ + θ f + A ) φ k , D φ k φ k 2 0 k =1 Mixed symmetry tensors in the worldline formalism
Introduction Gauge Holonomy Non-Abelian interactions Results Conclusion Generalisation Grassmann colour fields Using our earlier results for each family the path integral evaluates to � ∞ � 2 π � F dT dθ k � 2 π � 2 π e − in k θ k � ω 2 T + ψ · ˙ � 1 − e − iθ j e iθ k � D ω D ψ e − 1 ˙ ψ × 2 0 T 0 0 k =1 j<k F � � � ) e iθ k +tr W ( ) e 2 iθ k + . . . +tr W ( . . ) e ( N − 1) iθ k +tr W ( · ) e iNθ k tr W ( · )+tr W ( k =1 We use this formula to project onto the representation with n k rows in each column: n F ... ...n 1 ... . . Ψ( x, ˜ φ ) ∼ . ... . ... � �� � F columns Mixed symmetry tensors in the worldline formalism
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