KHRehren Wuppertal, June 2018 Quantum stress-energy tensors 1 / 28 Quantum stress-energy tensors without action functional Karl-Henning Rehren Universit¨ at G¨ ottingen, Institut f¨ ur Theoretische Physik LQP 42, Wuppertal, June 23, 2018
KHRehren Wuppertal, June 2018 Quantum stress-energy tensors 2 / 28 Abstract The stress-energy tensor describes (among other) the coupling of matter to gravity. Classical stress-energy tensors are usually derived by variations of the action . While already for Maxwell, the canonical formula gives an asymmetric and non-gauge invariant result, the Hilbert prescription (by variation of the metric) gives the correct symmetric and gauge-invariant stress-energy tensor. For higher spins, the problems become more severe because the action has to take care of manifold constraints. In the quantum case, additional problems of indefinite metric arise on top, which lead to famous no-go results. I present an alternative approach that allows to construct higher-spin stress-energy tensors intrinsically via the Wigner representation , without reference to an action functional. The method also applies to infinite-spin representations.
KHRehren Wuppertal, June 2018 Quantum stress-energy tensors 3 / 28 The action principle The action principle is an extremely successful paradigm in classical mechanics and classical field theory. Invariants → field equations It became most influential for modern QFT (eg, via the path integral), and is often regarded as fundamental. Yet, . . .
KHRehren Wuppertal, June 2018 Quantum stress-energy tensors 4 / 28 I want to shed some vinegar over this beautiful picture
KHRehren Wuppertal, June 2018 Quantum stress-energy tensors 5 / 28 Higher spin: the trouble starts Covariant fields of spin s ≥ 1 have more components than physical degrees of freedom. Need for kinematical and dynamical constraints . ( → Gauge symmetry and Noether’s Second Theorem) Massive, spin one: ! ∂ µ A µ = 0 . Massive, spin two: ! ! ∂ µ A µν η µν A µν = 0 , = 0 . Massless, spin one: ! ∂ µ A µ = 0 + gauge invariance . . . .
KHRehren Wuppertal, June 2018 Quantum stress-energy tensors 6 / 28 Fronsdal 1978 ( Fierz-Pauli 1939): Spin ≥ 2 requires at least s − 1 auxiliary fields . The case s = 2 is possible with A = η µν A µν as unique auxiliary field: 4 ∂ κ A ∂ κ A − m 2 L = 1 4 F [ µν ] κ F [ µν ] κ − 1 2 ( ∂ A ) κ ( ∂ A ) κ − 1 2 ( A νκ A νκ − A 2 ) (where F [ µν ] κ = ∂ µ A νκ − ∂ ν A µκ , ( ∂ A ) κ = ∂ µ A µκ ), but this pattern ceases at s > 2.
KHRehren Wuppertal, June 2018 Quantum stress-energy tensors 7 / 28 Quantization: the trouble continues Canonical quantization fails because some components have no canonical conjugate momentum . Needs “gauge fixing”. Classical Lagrangean not unique : adding a total divergence to � L preserves S = L , and hence the Euler-Lagrange equations of motion; but may change the path integral. More drastic case: Nambu-Goto string vs Polyakov string (induced metric vs worldsheet metric) yield inequivalent quantization . Still results in indefinite “canonical” Hilbert spaces. The latter issue has no classical analogue. Needs “ghosts” and BRST. Even with such an arsenal of elaborate tricks: A queasy feeling remains.
KHRehren Wuppertal, June 2018 Quantum stress-energy tensors 8 / 28 Fortunately, one can construct local free fields of any mass and any finite spin directly on the physical Fock space ( Weinberg 1964), based on the underlying positive-energy representations of the Poincar´ e group ( Wigner 1939). Crucial ingredient: “intertwiners” to mediate between the covariance of one-particle states and the covariance of fields. Perturbative QFT along the lines of Bogoliubov and of Glaser-Epstein starts from free fields (no need of a “free Lagrangian”). The “interacting part of the Lagrangian” is just a linear space (of couplings) on which the renormalization group acts (by readjusting the coefficients).
KHRehren Wuppertal, June 2018 Quantum stress-energy tensors 9 / 28 Stress-energy tensor: more trouble Classical Maxwell: canonical stress-energy tensor (SET) is neither symmetric nor gauge-invariant. Can be fixed “by hand”. General: many ambiguities ( Belinfante 1940) Modern attitude: Hilbert SET via variation of a generally covariant action by the metric. Automatically symmetric and gauge invariant.
KHRehren Wuppertal, June 2018 Quantum stress-energy tensors 10 / 28 Quantum stress-energy tensor: yet more trouble Weinberg-Witten theorem (1980): For s > 1, a local covariant stress-energy tensor on the physical Hilbert space does not exist (even for free fields). (Conflict between covariant transformation laws of one-particle states and the purported SET). Then, how does massless quantum matter couple to the gravitational field?
KHRehren Wuppertal, June 2018 Quantum stress-energy tensors 11 / 28 Reminder: Wigner quantization p 0 ∈ Spec( P ), p = B p p 0 . unirep d of “little group” Stab( p 0 ) ⊂ L ↑ + induces one-particle unirep U 1 of P ↑ + (Mackey). Second quantization U = Γ( U 1 ) on Fock space H = Γ( H 1 ). Creation and annihilation operators a ( ∗ ) m ( p ) on H . m ( p ) U (Λ) ∗ = a ∗ Adjoint action U (Λ) a ∗ n (Λ p ) d ( W Λ , p ) nm , with the “Wigner rotations” W Λ , p = B − 1 Λ p Λ B p ∈ Stab( p 0 ). Free quantum fields � d µ ( p ) e ipx u M , m ( p ) a ∗ φ M ( x ) = m ( p ) + h . c .
KHRehren Wuppertal, June 2018 Quantum stress-energy tensors 12 / 28 φ M ( x ) transform covariantly : U (Λ) φ M ( x ) U (Λ) ∗ = D (Λ − 1 ) N M φ N (Λ x ) iff the intertwiner functions u M , m ( p ) satisfy u (Λ p ) = ( D (Λ) ⊗ d ( W Λ , p )) u ( p ) . φ M ( x ) are local if u M ′ , m ( p ) u M , m ( p ) are polynomial functions.
KHRehren Wuppertal, June 2018 Quantum stress-energy tensors 13 / 28 KHR: JHEP 11 (2017) 130
KHRehren Wuppertal, June 2018 Quantum stress-energy tensors 14 / 28 Quantum SET: Roadmap Want to write � d 3 xT µ 0 ( t , x ) P µ = � d 3 x ( x µ T ν 0 ( t , x ) − x ν T µ 0 ( t , x )) M µν = with a symmetric and conserved tensor of local quantum fields. � d µ ( p ) p µ a ∗ Rewrite P µ = i ( p ) a i ( p ) as �� m ( p 1 )( p 1 + p 2 ) µ d µ ( p 1 ) d µ ( p 2 ) a ∗ ( p 1 + p 2 ) 0 δ ( � p 1 − � p 2 ) δ mn a n ( p 2 ) . 2 � 1 d 3 x e − i ( � p 1 − � p 2 ) � x . Write δ ( p 1 − p 2 ) = 2 π
KHRehren Wuppertal, June 2018 Quantum stress-energy tensors 15 / 28 Find a decomposition of unity δ mn = g MN u Mm ( p ) u Nn ( p ) . Rearrange � � � P µ = − 1 d µ ( p 1 ) e ip 1 x u Mi ( p 1 ) a i ( p 1 ) ∗ � 2 g MN d 3 x · � � � ↔ ↔ d µ ( p 2 ) e − ip 2 x u Nj ( p 2 ) a i ( p 2 ) . ∂ µ ∂ 0
KHRehren Wuppertal, June 2018 Quantum stress-energy tensors 16 / 28 Symmetrize 1 ↔ 2 ⇒ � � P µ = − 1 ↔ ↔ ∂ 0 φ N : ( x ) ! 4 g MN d 3 x : φ M d 3 x T µ 0 ( x ) . = ∂ µ Similar (but more laborious) for Lorentz generators � M µν = 1 ↔ d µ ( p ) a ∗ i ( p )( p ∧ ∂ p ) µν a i ( p ) 2 � � � ! d 3 x = x µ T ν 0 ( x ) − x ν T µ 0 ( x ) . Conclude T µν ( x ) = − 1 ↔ ↔ 4 g MN : φ M ∂ ν φ N : ( x ) + ∂ κ [∆ T κ µν ] . ∂ µ Derivative terms needed to accomodate the Lorentz generators.
KHRehren Wuppertal, June 2018 Quantum stress-energy tensors 17 / 28 Massive vs massless The main issue in this program is to find a complete system of “localizing intertwiners” (ie. defining local quantum fields) satisfying δ mn = g MN u Mm ( p ) u Nn ( p ) . In the massive case, this is possible for any spin, giving SETs originally found by Fierz (1939) (without the derivative terms). In the massless case, there is an obstruction .
KHRehren Wuppertal, June 2018 Quantum stress-energy tensors 18 / 28 Reminder: The Maxwell case The origin of the obstruction is well known in the Maxwell case: A local intertwiner for a vector potential (transforming in D (Λ) = Λ) does not exist ( Weinberg 1964). The best one can achieve, is U (Λ) A µ ( x ) U (Λ) ∗ = (Λ − 1 ) ν � A ν (Λ x ) + ∂ ν X ] , µ ie the vector potential transforms as a vector up to an operator-valued gauge transformation. Of course, F µν transforms correctly as a tensor. The gauge-invariant Maxwell SET is quadratic in F µν .
KHRehren Wuppertal, June 2018 Quantum stress-energy tensors 19 / 28 Higher helicity The same obstruction (no local potentials, only local field strengths) occurs with higher helicity h , except that the local and covariant field strengths F involve more derivatives , so that quadratic expressions in F are incompatible with the scaling dimension of the SET. To build a SET, one has to work with (first derivatives of) potentials, that are either not exactly covariant or not exactly local .
KHRehren Wuppertal, June 2018 Quantum stress-energy tensors 20 / 28 String-localized fields Mund-Schroer-Yngvason (2004): Integrating the Maxwell field strength along a “string” extending from x to ∞ , yields a “string-localized potential” � ∞ ds F µν ( x + se ) e ν . A µ ( e , x ) := 0 Indeed, one has ∂ µ A ν ( e , x ) − ∂ ν A µ ( e , x ) = F µν ( x ), and the variation of A ( e , x ) wrt the direction e is a gradient. For fixed e , this is just the “axial gauge” choice of potential. But A µ ( e , x ) coexist for all e on the Wigner Hilbert space.
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