Quantum Energy Inequali1es

Transcription

1 Quantum Energy Inequali1es Elisa Ferreira PHYS 731 Ian Morrison Ref: Lectures on Quantum Energy Inequali1es, C. Fewster

2 Introduc1on and Mo1va1on Classical energy condi1ons (CEC) of GR- WEC, NEC, DEC and SEC have central role. Obeyed by (almost) all classical theories. Introduced to model maxer distribu1ons in discussions such as singularity theorems. QFT, most successful theory describing maxer, is incompa1ble with the EC. Ques1ons: 1. Validity of sing. Theorems for realis1c maxer? 2. Can quantum fields support/allow exo1c space1me geometries such as wormhole, 1me machines,? Require stress-energy tensor viola1ng EC but are solu1ons of Einsteins equa1ons. Quantum energy inequali1es to the rescue: condi1ons within QFT that constraint the extend to which the CEC are violated.

3 Review: Classical Energy Condi1ons Why we need Ecs? R ab 1 2 Rg ab = T ab formal condensa/ons of all things whose comprehension in the sense of field theory is s/ll problema/c. A. Einstein Low grade of rhs: Only place in physics that studies T ab. All other fields of physics study only differences between stress-energy tensors. EE have NO predic1ve power example, every Lorentzian smooth st solves EE for a suitable (allows to construct wormholes, 1me machines, ) T ab T ab But, what sort of are physically reasonable? Need principles to determine it. CEC are an axempt to build such principles based on physical and mathema&cal arguments. * If maxer sa1sfies EC + EE holds = corresponding condi1ons in the geometry! (ex.: NEC implies R for all null u a ab u a u b 0.) Those EC enforce various focusing behaviors for congruence of geodesics.

4 Review: Classical Energy Condi1ons Weak Energy Condi1on (WEC) T ab u a u b 0 for all 1melike u a All observers see a non-nega/ve energy density. Null Energy Condi1on (NEC) Perfect fluid: T ab = (ρ + p)u a u b pg ab ρ 0 and ρ+p 0 T ab n a n b 0 for all null n a ρ+p 0 Dominant Energy Condi1on (DEC) T ab u a v b 0 for all future-poin1ng u a and v b ρ p All observers see a causal flux of energy momentum. Strong Energy Condi1on (SEC) T ab u a u b 1 2 gab T ab 0 for all 1melike unit u a DEC WEC NEC SEC NEC ρ 0 and ρ+3p 0

5 Viola1on of EC in QFT Lets use as an example Φ 2. Classically, Canonically quan1zed scalar field: Define φ 2 (x) Ω = lim x y φ(x) = φ 2 (x) = limφ(x)φ(y) x y φ 2 0. d d 1 k 1 "e " (2π $ ) # 3 # ikx a k + h.c. $ % $ 2ω % d! k φ(x)φ(y) Ω = lim d k! 1 d k! 2 e i(k 1x k 2 y) x y a k1 a + k 2 = Ω Composite operator!! d d 1 k 1 (2π ) 3 2ω =!!! To remove/renormalize these infini1es, those composite operators are defined normal ordered wrt a state (so, they are not unique), and form a Wick square :φ 2 (x) : :φ 2 (x) := d k! 1 d k! 2 {e i(k 1 k 2 )x a k1 a k2 + e i(k 1 k )x 2 a + k 1 a + k 2 + e i(k 1 k )x 2 a + k 1 a k2 + e i(k 1 k )x 2 a + k 2 a k1 } :φ 2 (x) : Ω = φ 2 (x) φ 2 (x) Ω

6 Viola1on of EC in QFT The smeared Wick square: :φ 2 [ f ]:= d 4 x :φ 2 (x) : f (x), f C 0 (R 4 ), f any test function on the ST. Applying in the vacuum state: :φ 2 [ f ]: Ω = d k! 1 d k! 2 ˆf (k1 + k 2 )a + k 1 a + k 2 Ω So, the norm : :φ 2 [ f ]: Ω 2 = Ω :φ 2 [ f ]: Ω = 0 d d 1 k 1 d d 1 k 1 1 ˆf (k1 + k (2π ) d 1 (2π ) d 1 2 ) 2 k 1 k 2 k 1 k 2 2ω 1 ω 2!# "# $ 0! 1 Unless f=0 Reeh-Schlieder theorem! :φ 2 (x) : has vanishing expecta1on value (in Ω) but it doesn t annihilate Ω!! :φ 2 (x) : has to have some nega1ve spectrum.

7 Viola1on of EC in QFT Assume the state: ψ α = cosα Ω + sinα :φ 2 [ f ]:Ω And evaluate the Wick square expecta1on value in this state: ψ α :φ 2 [ f ]: ψ α = sin2α + sin 2 α Ω :φ 2 [ f ]: 3 Ω 2α +O(α 2 ) If α < 0, :φ 2 [ f ]: ψα can be negative! Even if f non-nega1ve. or Each individual measurement of A in the vacuum is a random variable with A Ω = 0

8 Viola1on of EC in QFT Another very important composite operator is the energy-mom. Tensor. This nega1vity can happen for T ab as well. Consider a classical scalar field ϕ with stress tensor: Energy density: Now, consider a massless quantum scalar field in a state

9 Nega1ve Energy in Physics Casimir Effect: (conducing) Parallel plates in vacuum T µν = C(z) L 4 T out µν = C 0 L 4 diag( 1,1,1, 3) diag( 1,1,1, 3) AXrac1ve force In the limit L->0 or L-> infinity, is effec1vely as having only one plate à C 1 =0 p = 3C 0 L 4 T in µν = C 1 L 4 diag( 1,1,1, 3) C 0 > 0 (for attractive force) C 1 C 0 C 1 L ρ = T 00 = C 0 < 0 WEC is violated!! 4 L Full computa1on: C 1 = 0, C 0 = π 2 / 720( ) Small!!! Energy condi1on always almost sa1sfied! Viola1ons are either small or short lived! (or require disparate scales, or highly non-iner1al mo1on, or large posi1ve energies somewhere in the system)

10 Nega1ve Energy in Physics Although EC almost holds, we cannot insist on pointwise EC in QFT. We can see that for states like ψ, that are superposi1on of the vacuum and two-par1cle states kk >, fringes are formed, reminescent of the interference paxern. Fringes are spacelike. Timelike observer alternates seeing + and values. Cannot stay in a + or fringe. Suggests seeking constraints on local averages of ρ along 1melike curves.

11 Quantum Energy Condi1ons Quantum à Semiclassical gravity Nega1vity of rhs may allow exo1c geometries. QEI come to the rescue!! But what are QEI?

12 Quantum Energy Conditions

13 Quantum Energy Conditions

14 Quantum Energy Conditions

15 Quantum Energy Conditions Global hyperbolic ST Let γ : R M be a smooth 1melike curve, with proper 1me parameteriza1on. We consider the quan1ty : (Qφ 2 ) : ωr à NO wrt Hadamard states ω R We want a lower bound to: Hadamard state Lets introduce a point split quan1ty: Where G R is the same quan1ty evaluated at w R, the reference state. G and G R are distribu1ons à is a smooth func1on, symmetric and with diagonal:

16 Quantum Energy Conditions Inser1ng a delta to unsplit the points Thinking of F as a distribu1on Real valued g and using the symmetry of F: (1) * Can be applied to complex g à apply the above arguments for the real and imaginary parts!

17 Quantum Energy Conditions Previous calcula1on relied in the fact the GR is posi1ve type à the integrand is pointwise posi1ve in alpha. We found a energy inequality for : (Qφ 2 ) : ωr. Depends only on ω R, g and γ Any other quan1ty can be expressed as a finite sum of such quan1ty. (ex.: minimally coupled scalar field) (1) Ques1ons remain: Is it legi1mate to restrict the differen1ated two-point func1on to the world-line, as we did in defining G? No for a null trajectory à No QEI in this case. Is the final integrand in (1) finite? It is possible to diverge! Need microlocal analysis to reach those answers!

18 Quantum Energy Conditions Remarks: 1. We could take averages of other classically posi1ve contrac1ons of T ab along 1melike curves, e.g., against null vector à QNEI 2. Variants of this average for Lorentzian submanifolds. 3. No such bounds exist for smearing over spacelike surfaces. 4. Above argument (and for energy density), relies on classical posi1vity of the quan1ty in ques1on. This allows for QEI for spin-1, Dirac fields,

19 Quantum Energy Conditions Dependence on the reference state:

20 n=4 Minkowski space for minimal coupling Iner1al trajectory: Q a par1al diff. operator (with constant real coefficients) à, p(k) a polynomial with. Minkowski vacuum as the reference state: NO is the usual in Minkowski space QFT. γ(t,x) is the trajectory. So, And the required quan1ty in (1) is:

21 n=4 Minkowski space for minimal coupling Deriving the QEI for the energy density. The classical energy density of a minimally coupled scalar field in Minkowski is Leads us to considering the diff. operator Changing variable from k to ω, and from

22 n=4 Minkowski space for minimal coupling Where: For m=0: State dependent State independent Scaling behavior: Consider QEI contain a lot of informa1on!!, then the bound is As τà 0, the expecta1on value or the expecta1on value of ρ at a point is unbounded from bellow. As τà infinity WEC holds in this average sense à AWEC

23 n=4 Minkowski space for minimal coupling Bounds on the dura&on of nega&ve energy: Suppose that some interval of 1me. For any for Integra1ng by parts twice, this yields, for all nonzero g: L 2 inner products Minimizing the lhs, we have the bound: For some 1me interval, the energy density must at some instant exceed C/τ 4.

24 n=4 Minkowski space for minimal coupling Quantum interest: The QEI can be understood as asser1ng that the diff. operator: Is posi1ve in any interval or the reals, with the func1on and first deriva1ves vanishing at any boundaries. This leads to substan1al restric1on to ρ. Example: ρ has an isolated pulse: ρ(t)=0 on [t 1 -τ 1, t 1 ] and [t 2, t 2+ τ 2 ], t 1 < t 2 G a smooth func1on =0, near t=1 and =1 near t=0:

25 n=4 Minkowski space for minimal coupling Quantum interest: This amounts to an Euler-Lagrange equa1on Nontrivial constraint where a pulse (of any shape) can be isolated if the integral is nega1ve. If ρ is compactly supported, it can only be compa1ble with the QEI restric1on if it has non-nega1ve integral (AWEC). If ρ compa1ble with QEI and ρ = 0 ρ = 0. Financial Analogy (Ford and Roman): nature allows you to borrow nega1ve energy density but you must repay it within a maximum loan 1me à quantum interest effect