Quantum Feld Theory for Leg Spnners 17/1/11 Lecture Notes 7: The Unruh Effect Lecturer: Prakash Panangaden Scrbe: Shane Mansfeld 1 Defnng the Vacuum Recall from the last lecture that choosng a complex structure defnes the postve and negatve frequency solutons of your theory. Ths choce s not unque and one needs some nput from the physcs to pck out the rght complex structure. We consder what happens when two observers have dfferent notons of postve and negatve energy frequences. Suppose we have two complete, orthonormal sets of of complex solutons to the Klen-Gordon equaton, {f } and {q }. By orthonormal we mean that the unon of the f and ther complex conugates satsfy the followng equatons f, f = f, f = δ. So the Klen-Gordon nner product only behaves lke an nner product when restrcted to postve frequency solutons. By complete we mean that they form a bass n the complex vector space of all solutons. If ˆφ(x s our feld operator, then the creaton and annhlaton operators wth respect to the f-bass are defned by ˆφ(x = ( â f + â f, and the vacuum s defned as the unque state that s klled by all the annhlaton operators: other: â =. Snce we are dealng wth complete sets, we can wrte the f and g n terms of each So, for example, α = g, f. g = (α f + β f f = ( α g + β g g = ( α f + β f f = ( α g + β g. 7-1
We could equally well expand the feld operator n the g-bass. Then we have (ˆb g + ˆb g, ˆφ(x = for some other creaton and annhlators ˆb and ˆb, and we also have another vacuum defned by ˆb =. The queston we are nterested n answerng s: what does a one vacuum look lke n the other bass? 2 Bogoloubov Transformatons It s a routne calculaton to fnd that â = [ ] α ˆb + β ˆb â = [ ] αˆb + β ˆb ˆb = [ ] α ˆb + β ˆb ˆb = ] [α ˆb + β ˆb. These are called the Bogolubov transformatons. Now we have two number operators: ˆN = â â ˆN = ˆb ˆb. So we can take one vacuum and wth t calculate the expectaton value of the other number operator: ˆN = β 2. We see that under the Bogoloubov transformatons the annhlaton operators pck up a creaton part, and the coeffcents gve rse to the non-zero rght-hand sde here. It s ths mxng of the creaton and annhlaton operators that s responsble for partcle creaton. The Boboloubov transformaton formalsm was frst used n order to descrbe partcle by Leonard Parker n the 196s. 3 Rndler Spacetme Recall that a vector feld s a Kllng feld f the Le dervatve of the metrc along that vector feld vanshes. In Mnkowsk spacetme we have the boost Kllng feld: t z + z t. The ntegral curves of ths Kllng feld are tmelke n the rght and left Rndler wedges (RRW and LRW, see fgure 1. So n these regons of the spacetme we could use these 7-2
Fgure 1: Worldlnes of unformly accelerated observers. curves to defne a tme coordnate. They curves are curves of constant acceleraton, so for a unformly acceleratng observer these are the natural tme coodnates. An acceleratng observer wll see the dagonals of the fgure as horzons. So for such an observer, the Rndler wedge s hs natural home. Note that each of the left and rght wedges s a globally hyperbolc statc spacetme n ts own rght. The coordnates of a unformly accelerated observer are the Rndler coordnates (ρ, η, x, y. These are related to the Mnkowsk coordnates (t, z, x, y by the transformatons t = ρ snh η, z = ρ cosh η. The lne element of the Rndler spacetme s ds 2 = ρ 2 dη dρ 2 dx 2 dy 2. Lnes of constant acceleraton are lnes of constant ρ, and the value of the acceleraton s ρ 1. In terms of the scaled coordnates (τ, η, x, y, where the acceleraton a s factored out: ρ = a 1 e aξ, η = aτ. ξ = corresponds to acceleraton a. We wll use τ to defne postve and negatve frequency, and thus the vacuum for an acceleratng observer. We wll consder the dfference between the vacuum of Mnkowsk coordnates, and that of Rndler coordnates. It turns out that, to the accelerated observer, the Mnkowsk background looks lke a thermal bath: ths s what s known as the Unruh effect or the 7-3
Fullng-Daves-Unruh effect. We derve ths for a two-dmensonal spacetme wth massless felds. Ths s a toy demonstraton of the Unruh effect. It holds true n the four dmensonal massve case, but the analyss s qute a bt longer. 4 Toy Demonstraton of the Unruh Effect We wrte the momentum operator n terms of rght-gong and left-gong parts; each wth ts own creaton and annhlaton operators. ˆΦ(t, z = dk 4πk (ˆb k e k(t z + ˆb k e k(t+z + ˆb k e k(t z + ˆb k e k(t+z. When we change coordnates from (t, z to (τ, ξ we get ( t 2 z 2 φ = ( τ 2 ξ 2 φ =. Ths s due to conformal nvarance. It does not happen for four dmensonal massve felds, and for that reason they are a bt more dffcult to deal wth. If we use u = τ + ξ v = τ ξ U = t + z V = τ + ξ then we can wrte the momentum n the RRW n terms of the other coordnates as dk ˆΦ ω (v = (α ωk R e kv b? + βωk R ekv b?, 4πk and smlarly for the LRW. To calculate αωk R and βr ωk s ust an ntegraton now. For example, πω αωk R = e ( 2a a ω ( 2π a Γ 1 ω. ωk k a The thng to note here s the exponental. It s exactly the form you get n the Boltzmann dstrbuton for blackbody radaton. so f From ˆb M = we have ω â R f = f(ω â R ω dω, ( â R ω e πω a âr ω M = ; where ω then M â R f âr f M = dω f(ω 2 e 2πω/a 1, whch s the spectrum of a black body wth temperature a. 7-4 f(ω dω = 1,
Fgure 2: The Unruh-Wald detector. The two-state quantum mechancal system s coupled to a Klen-Gordon feld, and has a transton probablty due to ths feld. In ths pcture, a partcle s the effect of a detector nteractng wth a feld. 5 The Rndler Observer An acceleratng observer would use Rndler coordnates. Unruh and Wald consdered what would happen f an acceleratng observer had a partcle detector. By a detector they meant a smple two-state quantum mechancal system (states,, coupled to a Klen-Gordon feld (see fgure 2. Absorbng a partcle from the feld, the quantum system could change state, n, n 1. They showed that such a state, f accelerated n Mnkowsk space, wll detect a partcle wth probablty gven by the black body spectrum. Ths rases another ssue: how a Mnkowsk observer nterprets the absorpton of a Rndler partcle. Unruh and Wald showed that the Mnkowsk observer nterprets ths as the emsson of a Mnkowsk partcle. The problem now s: who thnks that the energy of the feld has ncreased or decreased? In fact, both observers wll agree that the energy n the feld has ncreased. It seems counter-ntutve n that the Rndler observer has absorbed a partcle from the feld, and yet the feld energy goes up. In general, however, when a partcle s absorbed from a heat bath, the energy goes up. The followng toy calculaton s a smple demonstraton of the fact. 7-5
Example Consder the state + 1 n n where n >> 1 s the number of quanta, each havng energy E. The expectaton value of energy s E E. If, however, you detect a partcle, then Energy (n 1E >> E. 6 Rndler Vacuum? An nterestng queston to consder s whether a Rndler observer can even prepare a vacuum. Of course, f not, then the observer can t prepare a pure state of any knd. Ths s a problem because n quantum nformaton theory, all of our protocols begn wth preparng pure states. If the Rndler vacuum state can be prepared, then whch has lower energy? Surely observers should agree on the orderng of energy levels, f not on the scalng. It has been calculated that the vacuum energy of the Rndler spacetme dverges as the horzons are approached. Ths may be taken as an ndcaton that the Rndler vacuum s unphyscal. However, much remans to be understood. 7-6