Mixed symmetry tensors in the worldline formalism James Edwards - - PowerPoint PPT Presentation

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

1

Introduction Motivation Worldline formalism

2

Non-Abelian interactions Gauge Holonomy Results Generalisation

3

Conclusion

Mixed symmetry tensors in the worldline formalism

Introduction Non-Abelian interactions Conclusion Motivation Worldline formalism

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

• f 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 Non-Abelian interactions Conclusion Motivation Worldline formalism

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 Non-Abelian interactions Conclusion Motivation Worldline formalism

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: Γ [A] = log

• D

¯ ΨΨ

• exp
• −
• d4x ¯

Ψ (γ · D − m) Ψ

• = −1

2Tr log

• (γ · D)2 + m2
• 1Nucl. Phys. B385

Mixed symmetry tensors in the worldline formalism

Introduction Non-Abelian interactions Conclusion Motivation Worldline formalism

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) Γ [A] ∝ ∞ dT T

• D (w, ψ) exp (−Spoint (w, ψ))W [A]

where Spoint = 1 2 1 ˙ ω2 T + ˙ ψ · ψ dτ and W [A] = trP

• exp
• iq
• dω · AAT A + iqT

2 1 dτ ψµF A

µνT Aψν

• Mixed symmetry tensors in the worldline formalism

Introduction Non-Abelian interactions Conclusion Gauge Holonomy Results Generalisation

The Wilson loop

W [A] = trP

• exp
• iq
• dω · AAT A + iq

2 1 dτ ψµF A

µνT Aψν

• How can we include gauge interactions and path ordering in the worldline

theory? Introduce “colour” fields ˜ φr and φr with Poisson brackets {˜ φr, φs}PB = −iδrs. Define RA ≡ ˜ φrT A

rsφs and note that {RA, RB}PB = if ABCRC.

We specify the dynamics of these new fields with the action 1 dτ ˜ φ (dτ + A) φ. 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 Non-Abelian interactions Conclusion Gauge Holonomy Results 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

• f the T R:

Ψ(x, ˜ φ) = Ψ(x) + Ψr1(x)˜ φr1 + Ψ[r1r2](x)˜ φr1 ˜ φr2 +· · ·+Ψ[r1r2..rN] ˜ φr1 ˜ φr2 ...˜ φrN Φ(x, ˜ φ) = Φ(x) + Φr1(x)˜ φr1 + Φ(r1r2)(x)˜ φr1 ˜ φr2 +· · ·+Φ(r1r2..rp) ˜ φr1 ˜ φr2 ...˜ φrp + ... 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. Z [A, θ] =

• D
• ˜

φ, φ

• exp
• −1

2 1 dτ ˜ φ (dτ + A + θ)) φ

• 2arXiv:0503.155[hep-th]

Mixed symmetry tensors in the worldline formalism

Introduction Non-Abelian interactions Conclusion Gauge Holonomy Results Generalisation

Wilson-loops

If the colour fields are taken to be Grassmann valued, we find ZN [A, θ] ∝ trW( · ) + trW( )e2iθ + trW( )e3iθ + . . . + trW( . . )e(N−1)iθ + trW( · )eNiθ . If the colour fields are instead bosonic, we have ZN [A, θ] ∝ trW( · ) + trW( )eiθ + trW( )e2iθ + trW( )e3iθ + . . . + trW( ·· )epiθ + . . . . Integrating against 2π

dθ 2πe−iθn picks out the representation with exactly

n fully (anti-)symmetrised indices.

Mixed symmetry tensors in the worldline formalism

Introduction Non-Abelian interactions Conclusion Gauge Holonomy Results 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: S[ω, ψ, ˜ φ, φ] = 1 2 2π dτ ˙ ω2 T + ψ · ˙ ψ + F

k=1

˜ φk

r ˙

φk

r + ˜

φk

rArsφk s

• .

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) ∼

• {n1,n2,...n

F }

. .

• nF

⊗ . . . ⊗ . .

• n2

⊗ . .

• n1

Φ(x, ˜ φ) ∼

• {n1,n2,...n

F }

··

• nF

⊗ . . . ⊗ ··

• n2

⊗ ··

• n1

.

Mixed symmetry tensors in the worldline formalism

Introduction Non-Abelian interactions Conclusion Gauge Holonomy Results 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 ajk(τ) for the generators of this symmetry group Ljk = ˜ φr

jφkr but only for k j

This partial gauging allows for the introduction of independent Chern-Simons terms fixing the occupation number of each family S = 1 dτ

• k

akk(τ)nk 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 Non-Abelian interactions Conclusion Gauge Holonomy Results Generalisation

Worldline theory

One can gauge fix by setting ajk = diag(θ1, θ2, . . . θF ) where the θk are moduli to be integrated over. The Faddeev-Popov determinant gives a measure for these moduli µ ({θk}) =

j<k µ ({θk, θj}) = j<k

• 1 − e−iθjeiθk

We use path integral quantisation on this gauge slice: ∞ dT T

• DωDψ e− 1

2

2π

˙ ω2 T +ψ· ˙

ψ F

• k=1

2π dθk 2π e−inkθkµ ({θk}) Z(F ) [A, {θk}] The partition function of the extended colour fields is Z(F ) [A, {θk}] =

F

• k=1
• D

˜ φkφk

• e− 1

2

2π dτ ˜ φk( ∂

∂τ +θf +A)φk, Mixed symmetry tensors in the worldline formalism

Introduction Non-Abelian interactions Conclusion Gauge Holonomy Results Generalisation

Grassmann colour fields

Using our earlier results for each family the path integral evaluates to ∞ dT T

• DωDψ e− 1

2

2π

˙ ω2 T +ψ· ˙

ψ F

• k=1

2π dθk 2π e−inkθk

j<k

• 1−e−iθjeiθk

×

F

• k=1
• trW( · )+trW(

)eiθk +trW( )e2iθk +. . .+trW( . . )e(N−1)iθk +trW( · )eiNθk

• We use this formula to project onto the representation with nk rows in

each column: Ψ(x, ˜ φ) ∼

nF ...

. .

...

...

...

...n1

. .

• F columns

Mixed symmetry tensors in the worldline formalism

Introduction Non-Abelian interactions Conclusion Gauge Holonomy Results Generalisation

Bosonic colour fields

For commuting colour fields we find ∞ dT T

• DωDψ e− 1

2

2π

˙ ω2 T +ψ· ˙

ψ F

• k=1

2π dθk 2π e−inkθk

j<k

• 1−e−iθjeiθk

×

F

• k=1
• trW( · ) + trW(

)eiθk + trW( )e2iθk + . . . + trW( ·· )epiθk + . . .

• This version projects instead onto the representation with nk columns in

each row: Φ(x, ˜ φ) ∼

nf

. . . .

n1

. .

...

...

...

. .             

F rows

Mixed symmetry tensors in the worldline formalism

Introduction Non-Abelian interactions Conclusion

Conclusion

We discussed worldline theories as a reformulation of QFT, describing interactions with the gauge field by introducing new colour degrees of freedom into the quantum theory.

1

These fields create the Hilbert space of the colour degrees of freedom, whose wavefunctions transform in a reducible representation.

2

Gauging a unitary symmetry allows us to project onto a single irreducible representation, which is fully (anti-)symmetric.

3

More general representations can be included by using multiple families of colour fields to produce tensor products representations.

4

A partial gauging of the resulting unitary symmetry picks out a single irreducible representation with the desired inter-family symmetries.

5

This allows the worldline formalism to be used for arbitrary matter multiplets coupled to a non-Abelian gauge field; this matter may be scalar, fermionic or higher spin.

Mixed symmetry tensors in the worldline formalism

Introduction Non-Abelian interactions Conclusion

Conclusion

We discussed worldline theories as a reformulation of QFT, describing interactions with the gauge field by introducing new colour degrees of freedom into the quantum theory.

1

These fields create the Hilbert space of the colour degrees of freedom, whose wavefunctions transform in a reducible representation.

2

Gauging a unitary symmetry allows us to project onto a single irreducible representation, which is fully (anti-)symmetric.

3

More general representations can be included by using multiple families of colour fields to produce tensor products representations.

4

A partial gauging of the resulting unitary symmetry picks out a single irreducible representation with the desired inter-family symmetries.

5

This allows the worldline formalism to be used for arbitrary matter multiplets coupled to a non-Abelian gauge field; this matter may be scalar, fermionic or higher spin. Thank you for your attention.

Mixed symmetry tensors in the worldline formalism