Hawking , the Unruh Effect and the Polarization of Electrons Jon Leinaas Institute of Physics, University of Oslo, Norway The polarization of electrons induced by quantum fluctuations of the field can be used as a robust radiation detector for the Unruh effect, where an idealized thermometer accelerated through vacuum registers a non-zero , thereby providing a laboratory probe for . In standard quantum field theories, An interesting feature of this theoretical the vacuum state is the building block effect is that the radiation is thermal, on which the full of states of the with the temperature given by system is based. It is the of the Hamiltonian, and corresponds to "empty space" in the conventional un­ where k is Boltzman's constant, G is derstanding of space and matter. How­ Newton's gravitational constant and M ever, in quantum theories, vacuum is is the mass of the . not completely free from structure. This Hawking derived this result by consi­ structure can be understood in terms of dering the continuation of field modes "quantum fluctuations" of the fields — through a collapsing massive body. fluctuations which are present even However, it seems clear that such an when there are no excitations (particles) effect is also present for an eternal present in the system. This vacuum ac­ Fig. 1 — Accelerated black hole, and the presence of radiation tivity is somewhat analogous to the (τη). Ordinary is hot with is then related to the problem of defi­ zero point fluctuations existing in a ma­ respect to the corresponding Hamiltonian, ning a stationary vacuum state in the terial system at zero temperature, fluc­ defined in either the " right Rindler wedge" gravitational field of the black hole. tuations which are, for example, respon­ (RRW) or in the "left Rindler wedge" (LRW). sible for helium remaining fluid even at A uniformly accelerated point detector, fol­ lowing a path with η fixed is heated due to Unruh Effect the lowest possible . interactions with vacuum fluctuations. Formally, there is a close resemblance Vacuum fluctuations can give rise to between the Hawking effect and the physical effects when the boundary Hawking Radiation presence of radiation quanta in an ordi­ conditions defining the vacuum are nary Minkowski vacuum when the lat­ changed. Thus, if conducting plates are The physical vacuum is influenced not only by physical boundaries, but ter is referred to the accelerated Rindler introduced into the electromagnetic coordinates. This resemblance is de­ vacuum, the fluctuations of the fields also by the presence of external fields. Thus, the vacuum fluctuations of char­ monstrated in a clear way by what is around the plates will be influenced by often referred to as the Unruh effect. the plates, and this gives rise to forces ged fields will be influenced by electric fields, and if these become sufficiently According to Unruh [3], an idealized between the plates (the well-known detector which is uniformly accelerated ). However, even without strong, the vacuum is no longer stable: the strong electromagnetic field gives through Minkowski vacuum will detect referring to changing boundary condi­ radiation with a thermal spectrum. The tions, there is a certain ambiguity in the rise to spontaneous pair creation. Strong gravitational fields may also give temperature formula has a form closely definition of the vacuum state which related to that for Hawking radiation, comes from the fact that the Hamilto­ rise to similar vacuum effects. As nian in relativistic theories is linked to shown by Hawking [2], the vacuum the choice of space-time coordinates. fields around a black hole are unstable Consequently, even when ordinary Min- and spontaneously emit quanta. with a as the acceleration of the detec- kowski-space quantum field theories are referred to non-inertial coordinates, the ordinary vacuum is no longer the ground state [1]. If a special choice of uniformly accelerated coordinates is made (Rindler coordinates) a new ground state can be found (Fig. 1). How­ ever, relative to this "accelerated va­ cuum", the ordinary Minkowski vacuum is not empty, but contains excitations (radiation). Professor J.M. Leinaas is with the Institute of Physics, University of Oslo, Norway. He studied at the University and spent a year at From the left, J.M. the State University of New York at Stony Leinaas, H. Wind, Brook, USA before taking up Fellowships Η. O veraas and at NORDITA, Oslo (1978/80) and CERN, John Bell at CERN Geneva (1981/82). in August 1987. 78 Europhys. News 22 (1991) tor (as measured in an inertial rest Fig. 2 — A simplified picture of frame). In the same way as the Hawking electrons in a magnetic field, temperature is extremely small for black where only the spin interaction holes of astronomical size, the Unruh with the field is considered. At temperature is extremely small for "nor­ zero temperature, all the particles sit in the ground state, with spin mal" accelerations. By inserting num­ antiparallel to the magnetic field. bers into the temperature formula one At non-zero temperatures, some realizes that even a modest temperature particles are excited to the upper of 1 K requires an acceleration of a = energy level, with spin parallel to 2.4 X 1020 m s-2 : an extremely robust the magnetic field. detector is clearly needed to withstand such accelerations. One may therefore wonder if this has to remain a purely Fig. 3 — Polarization curves for theoretical effect, or whether the va­ electrons circulating in a magne­ cuum temperature may be seen in real tic field as functions of the gyro- experiments. magnetic factor g. A is the curve found using a naive application of Polarized Electrons Unruh's temperature formula : B A few years ago, as a Fellow at CERN, is the curve for a more detailed I discussed these problems with John analysis. Bell and we became interested In exami­ ning matters a little further. We were led to consider elementary particles as pos­ sible radiation detectors, since other physical systems hardly could be suffi­ ciently robust at the required accelera­ charge. An ensemble of such electrons account of the Thomas precession of tions. As suggested by John Bell, one at zero temperature will be fully polari­ the electron in the definition of the spin might ask whether the temperature zed in the direction opposite to that of Hamiltonian would shift the curve along effect could be related to a polarization the magnetic field since they will all sit the g-axis, but with 2 units rather than effect which was known to exist for In the ground state (Fig. 2). However, for with about 1.2 which seems to be more electrons circulating in a magnetic field. a non-zero temperature T there is a correct. It had already been established that certain probability for excitation to the Another indication that the main part electrons in a storage ring polarized upper energy level. The ratio R between of the complication relates to a spin spontaneously, but not completely [4] : the occupation of the corresponding effect is seen if one considers a modi­ the maximum polarization had been spin levels is then fied problem where the electromagnetic found to be P = 0.92. Could the depar­ field is replaced by a scalar radiation ture from full polarization be interpreted field. If a two-level system is coupled as due to heating of the electrons cau­ linearly to this field, and the occupation sed by their acceleration ? ratio R is calculated as a function of the As a result of our investigation, the energy splitting for circular motion, then answer to this question was a qualified the deviation from the result found affirmative. The effect was indeed of using the Unruh formula is very small the same type, but there were compli­ [page 72, 7]. cations owing to the fact that the elec­ If this naive picture is applied to elec­ trons were circulating instead of accele­ trons circulating in a magnetic field and Exact Analysis rating linearly ; Thomas precession was Unruh's temperature formula is used, In a more serious treatment of the another complicating factor. I will brief­ one finds electrons circulating in a storage ring, ly outline how the polarization effect we introduced the Hamiltonian defined could be related to the Unruh effect and in an accelerating coordinate frame, also point to the complications (refer to moving along the classical path of a cir­ [page 72, 7] for more details). where S' = yB and S' = βγβ are the culating electron. With rotationally inva­ magnetic and electric fields in the rest riant external fields, this Hamiltonian is Polarization and the Unruh Effect frame of the electrons, β = v/c close to time-independent, and the situation is Electrons in a magnetic field B can 1 for highly relativistic electrons and stationary as seen from the moving naively be thought of as a thermometer γ = (1 — β2)-1/2. For g = 2 this gives P electron. The Hamiltonian has the form in the following way: assume for the = 0.996 as compared with P = 0.924 (1) as before, but with a somewhat dif­ moment that a particle's position is found in earlier QED calculations. The ferent form for the vector Ǡ, which unimportant and that only the spin de­ naive picture is clearly too simple to give now depends both on the classical ex­ gree of freedom is of interest. The spin the correct result. However, it is interes­ ternal fields and the quantum radiation Hamiltonian has the form ting to compare the results for g not field, as well as on the position vector equal to 2. As shown in Fig. 3, the main and velocity of the electron-variables difference between the function P(g) describing fluctuations around the clas­ with ω as the Pauli spin matrices and σ given above and that of the more com­ sical path. as the vector (eg/2mc)B where g is the plete QED calculation [5] is a shift of the The classical fields define the energy gyromagnetic factor, which is close to curve along the g-axis. Such a shift is levels of the spin Hamiltonian, and the 2 for electrons, and — e the electron in fact not totally unexpected. Taking quantum fluctuations determine the Europhys. News 22 (1991) 79 excitation probabilities. In our approach, of along the classical orbit. For f(g) = 0, character. This is not the case for the these excitations come about in two the result found in this way for the "circular Unruh effect” where the exci­ ways: the first is through the direct transverse polarization of the electrons tation spectrum depends on characte­ coupling of the spin vector to the quan­ agrees with the earlier result [5]. ristics of the detector. Using circulating tized electromagnetic field; the other An interesting additional observation electrons, the effects of Thomas pre­ is through fluctuations in the particle concerns the resonance factor f(g). In cession are important and fluctuations orbit. It is the vertical fluctuations (in our approach one sees that the reso­ in the orbit cannot be neglected. the direction of the magnetic field) nance does not have a purely depolari­ which induce spin transitions. Through zing effect. As a function of γ, the pola­ ADDITIONAL REFERENCES the Heisenberg equations of motion, rization close to the point of resonance [1] Fulling S.A., Phys. Rev. D 7 (1973) 2850. these fluctuations in turn can be related initially decreases, from 0.92 to -0.17, [2] Hawking S., Nature 248 (1974) 30; to the quantum fluctuations in the but then increases to 0.99 before finally Commun. Math. Phys. 43 (1975) 199. electromagnetic fields. The field fluctu­ decreasing again to 0.92. It is thus pos­ [3] Unruh W.G., Phys. Rev. D 14 (1976) 870. ations build up the fluctuations in the sible, at least in principle, to exceed the [4] Sokolov A.A. and Ternov I.M., Dokl. orbit and radiation damping prevents ''maximum'' value 0.92 close to the Akad. Nauk. SSR 153 (1963) 1052. them from growing indefinitely. resonance. [5] Derbenev Ya.S. and Kondratenko A.M., Zh. Exp. Teor. Fiz. 64 (1973) 1918 ; Jackson The part of the fluctuations in the Conclusions J.D., Rev. Mod. Phys. 48 (1976) 417. vector ώ, which is responsible for the Returning to the question of the rela­ spin excitations, can then be written in tion between the polarization effect and the simple form the Unruh effect one sees, as already stated, that they are of a similar nature. John Bell Symposium Transitions between spin levels are A Symposium on Quantum Phy­ where B'q and E'qz are the free quantum caused by quantum fluctuations of the sics, in memory of John Stewart Bell fields in the rest frame of the electron vacuum fields along the electron orbit, who died on 1 October 1990, will be and er is a unit vector in the radial direc­ in the same way as field fluctuations in­ held at CERN on 2-3 May. The pro­ tion. The resonance factor f(g) is pro­ duce transitions in a linearly accelerated gramme includes talks by A. Aspect, portional to g-2 : being a function of γ detector. In this sense, both the elec­ H. Rauch, A.J. Leggett, K. Gottfried, it is essentially equal to 0 except in a A. Shimony, G.C. Ghirardi, J. Leinaas trons and the detector are heated by the and R. Jackiw. Further information: narrow interval where there is reso­ accelerating motion through a vacuum. Jeanne Rostant, Theory Division, nance between the vertical fluctuations However, there is also an essential CERN, CH-1211 Genève 23 (Tel./Fax: and the spin motion. The transition pro­ difference between the two cases. For + 41 (22) 767 42 22 / 39 14; Bitnet: babilities between the spin levels now uniform linear acceleration, the excita­ rostant @ cernvm.cern.ch). are determined by correlation functions tion spectrum has a universal thermal World Lab Fosters Third World Excellence Establishing a "new equilibrium" in the Science for Peace Prize (see Europhysics distribution of knowledge between east News 21 (1990) 219). Housed in two con­ and west, north and south can provide a verted monasteries and a former convent "new way out" for undeveloped coun­ in the famous medieval hill-top town, the tries by allowing the creation of scientifi­ Centre has hosted some 100 schools and cally based cultures which have the capa­ 480 courses attended by 44000 parti­ city to generate wealth from limited cipants from 103 countries, a great de­ material resources. But one has to act monstration of "international collabora­ directly because if one "applies European tion without ideological position". Erice Professor S.P. Kapitza (left) receiving logic", this process would never have also gives its name to the world renowned on behalf of his father Academician started even in our "corner of the planet". Statement written by P.A.M. Dirac, P.L. P.L. Kapitza the first Scienza per la Action must also span all disciplines Kapitza and A. Zichichi whose 10000 Pace Prize from Professor A. Zichichi since "only a strong civilization creates signatories declare themselves morally last November in Erice, Sicily. science". Most important of all, expe­ bound to work purely in the interests of The other winners were P.A.M. Dirac rience in creating the great laboratories of science — a "new rôle for science". and A.D. Sacharov, E. Teller and V.F. Europe implies that solutions will mostly Weisskopf. spring from initiatives by individuals. World Laboratory These remarks by Professor Antonino The World Lab builds upon the Erice Zichichi help one understand the princi­ principles through collaborative projects in tacts so that distinguished colleagues can ples which guide the International Centre science and technology with existing help identify unique projects. The result is for Science Culture (ICSC), better known institutions. The Lab's operating budget that Professor T.D. Lee of Columbia Uni­ as the World Laboratory. Founded by comes largely from Italy, the USSR and versity was instrumental in setting up in Professor Zichichi, its President, in 1986, China, but this does not imply that it is just 1986, Beijing's Center for Advanced the World Lab works to extend the Erice another funding agency. Each beneficiary Science and Technology of which he is the heritage in improving the training of, and country in fact handles local projects, thus Director. The Center is a branch laboratory opportunities for, scientists based in requiring only a very small administrative established with governmental agreement poorer countries. Erice is the home of the team at the headquarters in Lausanne, that ensures independence and effecti­ Ettore Majorana Centre for Scientific Switzerland. veness in running projects in China. It Culture that celebrated its 25th Anniver­ The Erice experience also provided a employs 50 scientists and 30 students sary last year with the first award of the wealth of essential experience and con­ and, aside from research in high energy 80 Europhys. News 22 (1991)