The ideas of particle physics
This is a brief extract from the book
By G.D. Coughlan and J.E. Dodd
ISBN 0 521 38677 2 pbk
ISBN 0 521 38506 7 hard back
Cambridge University Press
… Recall that Einstein’s equations (Section 5.2.1) relate the geometry of space-time (which is actually what we call gravity) to mass and energy (which are sources of gravity). It seems natural, therefore, to reinterpret space-time in such a way that the gravitational waves (which are a form of energy and hence a source of gravity) are included with the sources on the right-hand side of Einstein’s equations. The geometry of space-time is therefore specified by the background space-time which remains on the left-hand side. Hence we may rewrite Einstein’s equations schematically as
(geometry of background space-time)=8πG*(mass and energy, including gravity waves)
So far, this is all classical. We shall now take account of the quantum nature of matter (including gravitational waves). That is, we shall consider the sources of gravity – matter and gravitational waves -as quantum fields. The background space-time will continue to be treated as classical. Then we have a semi-classical (and semi-quantum) theory, which might be regarded as a starting point for a full quantum theory of gravity. However, this theory leads to Feynman diagrams which are infinite and, in contrast to QED, cannot be renormalised (see Section 4.8). That the theory is not renormalisable can be traced to the fact that the gravitational coupling, G, is not dimensionless. Nevertheless, if quantum gravitational effects are limited to no more than one loop involving virtual gravitons (Figure 40. 1), then this truncated theory provides a useful framework for many calculations. This is analogous to Fermi’s current-current theory, which, despite being unrenormalisable, gives an adequate description of the weak interactions at low energies. Of course, it ignores higher-order quantum gravity effects, but this should be a valid approximation in most cases provided the relevant length and time scales are much greater than the Planck values.
Let us summarise the picture we have built up. The graviton is a massless spin-2 particle which is the quantum of the gravitational field, and hence mediator of the force of gravity. It propagates, along with other quantum fields, in a classical background space-time which satisfies Einstein’s equations. Furthermore, as general relativity interprets gravity in terms of space-time geometry, gravitons carry with them small disturbances to the classical background space-time. Despite being an incomplete formulation of quantum gravity, this approach has provided important insights and yielded many interesting results which are expected to form part of the full theory. Examples include the creation of particles in curved space-time, and Stephen Hawking’s celebrated discovery that black holes radiate particles.
40.3 Particle creation in curved space-time
Particle creation in curved space-time is an interesting illustration of the effects that gravity can have on quantum fields. In particular, particles can be ‘created’ by a gravitational field that is changing with time, such as the gravitational field of an expanding universe. Recall our discussion in Section 4.4, in which we considered quantum fields in terms of an infinite number of harmonic oscillators (masses attached to springs). We imagined harmonic oscillators at every point in space-time. Now, when the space-time curvature changes as it does in an expanding universe, this is equivalent to effecting a change in the physical properties of the oscillators. Suppose an oscillator is initially in its ground state (i.e. state of minimum energy), undergoing zero-point oscillations (see Section 4.9). Then if one of its physical properties (e.g. the mass, or the stiffness of the spring) is changed, the zero-point oscillations must readjust themselves. After this readjustment is complete, there is a non-zero probability that the oscillator is no longer in its ground state, but in a state of excited vibration corresponding to a particle.
As one might expect, particle production is greatest where the curvature is greatest and changing most rapidly. Moreover, the lightest particles are produced most easily, as they require less energy. But the energy of the created particles does not appear out of nothing: it comes from the gravitational field, that is, from the curvature of space-time itself.
Studies of particle production in various space-times have revealed that the concept of a ‘particle’ is not universal. It is observer-dependent. Particles may be registered by certain particle detectors and not by others. In particular, the state of motion of a particle detector can have a bearing on whether or not particles are observed. For example, even in flat space-time, a detector which is accelerating detects particles, even when an identical unaccelerating detector does not. In fact, a uniformly accelerated observer detects a thermal distribution of particles at a temperature of
where a is the acceleration and kb is Boltzmann’s constant. This should not be too surprising. By the principle of equivalence, the effects of acceleration and gravity are identical. So, it is as though the accelerating observer is sitting in a gravitational field, the curvature of which modifies the oscillators associated with any quantum field. Hence, what this observer calls a zero-point oscillation is completely different from what an unaccelerated observer calls a zero-point oscillation. In particular, zero-point oscillations for the latter correspond to excited states of vibration (i.e. particles) for the former.
40.4 Hawking radiation from black holes
Black holes are massive classical objects predicted by the theory of general relativity. A black hole can be defined as a region of space-time from which nothing – not even light – can escape. Consequently, they look completely black and can only be observed by virtue of the effects of their strong gravitational fields on neighbouring matter. The boundary of a (non-rotating) black hole of mass M is a spherical surface called the event horizon, situated at a radius of
This is called the Schwartzchild radius.
Black holes are believed to be the final evolutionary stage in the life of very massive stars. As an example, imagine a very massive star, many times the mass of the sun. For the first billion or so years of its life, this star shines brightly, with heat and light generated by nuclear fusion reactions. The energy released in the fusion processes creates sufficient pressure to stop the star from collapsing under its own gravity. This is a situation of hydrostatic equilibrium in which the inward gravitational force is exactly balanced by the outward pressure. However, once the star exhausts all its nuclear fuel, the pressure cannot be maintained and surrenders to gravity: the star collapses. During collapse, the gravitational field at the star’s surface increases enormously. Once the radius has fallen to the Schwartzchild value, it is so strong that even light cannot escape: a black hole has been born.
For stars 10 times more massive than the sun, the Schwartzchild radius is approximately 30 km. We now have much observational evidence to support the belief that black holes of about this size exist in binary star systems in which a visible star orbits an unseen companion (a black hole?). Much more massive black holes – 108 times the mass of the sun and with a Schwartzchild radius of 300 million km – may act as the power source at the centre of quasars. There is also the possibility that black holes very much lighter than the sun may also exist. Such black holes could not have formed by gravitational collapse – they are much too light. However, they might have formed under the very high temperatures and pressures prevailing in the early universe, shortly after the Big Bang. Such a ‘primordial’ black hole with the mass of a billion tonnes would have a Schwartzchild radius of about 10-13 cm, the size of an atomic nucleus! This is small enough to suggest that quantum-gravitational effects associated with primordial black holes may be significant – as indeed they are.
40.4.1 Entropy and thermodynamics
When matter falls into a black hole, its mass increases and, as a consequence of equation (40.1), so does the Schwartzchild radius. Since nothing can escape from beyond the event horizon, its surface area cannot decrease. This is reminiscent of a thermodynamic quantity called entropy (see box, p. 186), which is a measure of the degree of disorder in a system, or equivalently, of the lack of knowledge of the state of the system. One of the most fundamental laws in physics, the second law of thermodynamics, states that entropy cannot decrease. This similarity was the first hint of a connection between black holes and thermodynamics. It led to the suggestion, in 1972, that the area of the event horizon is a measure of a black hole’s entropy. As matter (which carries entropy) falls into a black hole, the horizon’s area increases by an amount which is, in fact, greater than the entropy of the in-falling matter. Not only does the black hole swallow up the entropy of the matter itself, but it also swallows up all information about the particular state it was in. For example, whether the matter was in the form of a star, or a cloud of gas, or lumps of rock makes no difference to the black hole. This information is lost and so the total entropy increases.
In fact, there is a mathematical theorem (called the ‘no-hair’ theorem) which states that, when a star collapses into a black hole, it quickly approaches a state specified by only three quantities: its mass, its angular momentum and its charge. No other details of the star survive. So, in the collapse we lose virtually all knowledge of the state of the star, and entropy increases by an enormous amount. For example, the entropy of a black hole of solar mass is about 1054 joules per degree, whereas the entropy of the sun is only about 1035 joules per degree.
However, there seemed to be a fatal flaw in the argument. If a black hole has entropy, then it ought to have a temperature, and hence it should emit radiation. But, according to classical physics, nothing at all can escape from a black hole. So, how can it possibly have a temperature?
40.4.2 Hawking radiation
Then, in 1974, came one of the most surprising and potentially one of the most fundamental results in modern physics. Stephen Hawking was investigating the behaviour of quantum fields in the vicinity of a black hole and found, to his surprise, that indeed black holes seem to emit particles at a steady rate, until they eventually evaporate together. Furthermore, the spectrum of the outgoing particles is precisely thermal: particles and radiation are emitted just as if the black hole were a hot body at a temperature of
In this expression kb is Boltzmann’s constant, M is the mass and TH is called the Hawking temperature. Hence, the identification of the surface area of a black hole’s event horizon with its entropy was complete. Table 40.2 gives the Hawking temperatures and approximate lifetimes of black holes of various sizes.
To appreciate how it is possible for black holes to radiate particles, recall the discussion of quantum vacuum in Section 4.9. According to the uncertainty principle, there are, even in empty space, zero-point quantum fluctuations associated with all quantum fields. These fluctuations may be thought of as pairs of virtual particles and antiparticles which constantly materialise out of the vacuum, briefly separate, and then annihilate. (See Section 33.l.)
In the vicinity of a black hole, the strong tidal forces may lead to one member of a virtual particle-antiparticle pair falling into the hole, leaving the other without a partner with which to annihilate. If the latter does not suffer the same fate as its partner, it becomes a real particle and appears to be emitted by the black hole (Figure 40.2). However, as this is a real process, energy and momentum must be strictly conserved. So, one can view the process in the following way. One member of the particle-antiparticle pair has positive energy and escapes to infinity, while the other enters the black hole with negative energy relative to infinity. The black hole absorbs the negative energy, thus reducing its mass by precisely the amount that the positive-energy particle carries away. The entropy of the black hole is decreased by the reduction in its mass, but this is more than compensated for by the entropy carried away by the emitted particle. Hence, overall, entropy is increased. Furthermore, we would expect the smaller the black hole, the stronger the tidal forces, the more likely it is for a virtual particle to fall into the hole, and therefore the greater the rate of emission. This is borne out in Table 40.2, where we see that smaller black holes have higher temperatures.
As a black hole radiates, it loses mass and, by equation (40.2), gets hotter. As it becomes hotter, it radiates more particles, loses more mass, etc. There comes a time in this runaway particle emission when the rate of mass loss is so great that the notions of thermal equilibrium and of a fixed background space-time are no longer valid. What happens after this time is not entirely certain. However, it is likely that in the final tenth of a second the black hole should release some 1023 joules (~= 1033 GeV) of energy – mostly in γ-ray photons. This is equivalent to a 106 -megatonne thermonuclear explosion!
One final consequence of black hole radiation is the violation of baryon number and other global quantum numbers. A black hole formed from collapse of a star forgets its baryon number and radiates a thermal spectrum of equal numbers of baryons and antibaryons.
40.5 Quantum cosmology
In the previous sections, we have seen how a semi-classical formulation of quantum gravity leads to interesting quantum effects when applied to black holes and expanding universes. It is apparent that quantum mechanics can modify the predictions of general relativity in very profound and surprising ways. Physicists have been encouraged by these results to conjecture that quantum mechanics might offer a way around the apparently unavoidable conclusion that the universe began from a singularity. Perhaps the true theory of quantum gravity will reveal how the Big-Bang singularity is ‘smoothed out’ by some quantum-gravitational uncertainty principle. If this is true, then space-time may have been perfectly smooth at the Big Bang and the laws of physics may have been valid even at the beginning of time (see Figure 40.3). In this section, we consider an approach which attempts to transcend the limitations of the semi-classical formulation and describe the large-scale quantum properties of the universe as a whole. This approach is based on Richard Feynman’s formulation of quantum mechanics in terms of a ‘sum over histories’.
40.5.1 The sum over histories
Let us put gravity aside for a moment to examine Feynman’s formulation. Consider a particle which moves from an initial point y at an initial time ti to a final point x at a final time tf. Classically, it follows a well-defined path, or history, determined by Newtonian physics. Quantum mechanics, however, tells us that this cannot be the case, since a precisely determined path is inconsistent with the uncertainty principle. We must, therefore, consider paths other than the classical one (Figure 40.4). According to Feynman, a particle does not necessarily move along the classical path, but may take any arbitrary path in space-time. So, a particle has an infinite number of possible histories. Corresponding to every path, Feynman assigned a quantum-mechanical amplitude which is a measure of how probable it is for that path to be taken. Then the probability that the particle passes through a particular point at a particular time is found by adding up the amplitudes associated with all the paths passing through that point. This is Feynman’s ‘sum over histories’.
The reader might be wondering what the connection is with the wavefunction of the particle. Well, we can find the particle’s wavefunction at a given time t0 by simply performing the sum over histories:
Ψ(x,t0)=Σ histories,H m[H]
where m[H] is the quantum-mechanical amplitude associated with a history H from an initial point y in the infinite past (ti = -∞) to the point x at a time tf = t0. This , is precisely equivalent to solving Schrödinger’s wave equation, and, hence, the two formulations of quantum mechanics are equivalent.
Because of technical problems in evaluating the sum over histories, quantum cosmologists usually work with ‘imaginary’ time. Imaginary time, τ, is related to ‘real’ time, t, by the formula τ= it, where i is the square root of -1. That is, i=√-1 , or i2 = 1. In imaginary time, the invariant space-time interval we met in Section 2.8.3, is given by
s2 = x2 + (cτ)2 (40.4)
since τ2=t2.. Note that there is now no minus sign distinguishing time from space – they both appear on an equal footing. So, time has become space-like. A space-time with this property is called Euclidean after Euclid, the father of geometry. (The above equation may remind the reader of the relationship between the squares of the sides of a right-angle triangle – a theorem accredited to Pythagoras, but based on Euclidean geometry.)
Hence, in imaginary time, four-dimensional space-time becomes four-dimensional Euclidean space.
Physicists hold greatly diverging views on the significance of imaginary time. Some argue that it is merely a mathematical trick, or a convenient tool devoid of any physical significance. Others suggest that imaginary time is the true physical quantity. In fact, the whole question of the meaning of time in quantum mechanics in general is very controversial. Many physicists hold that time is only a semi-classical concept and cannot – even in principle – be extended to the domain of quantum gravity. In this domain, imaginary time may perhaps be the appropriate concept.
40.5.2 The wavefunction of the universe
In quantum cosmology, the above ideas are applied not to a particle, but to the universe as a whole. Working in curved Euclidean space (i.e. with imaginary time), the probability that the universe has a particular space-time geometry at a particular time is given by the sum over all possible histories of the universe which lead to that geometry. The classical path, or classical history, is the one specified by Einstein’s equations, in the same way that the classical path of a particle is that given by Newton’s equations.
Just as one can consider the wavefunction of a particle, so can one also consider the wavefunction of the entire universe, an object which completely specifies its quantum state. (There are a host of interpretational and philosophical difficulties associated with this concept, which take us well into the realm of metaphysics and beyond the scope of this discussion.) The wavefunction of the universe is also given by a sum over histories similar to equation (40.3), and satisfies a Schrödinger-like wave equation which is called the Wheeler-De Witt equation. This equation is, like the sum over cosmic histories, extremely complicated. Every solution to this equation describes a universe. However, we are interested in one particular solution – the one corresponding to our universe – and we must impose suitable boundary conditions on the wavefunction to obtain this solution. That is, to obtain the wavefunction for our universe, we must impose boundary conditions describing its initial state. Unfortunately however, the appropriate boundary conditions are not known.
One proposal, put forward by Stephen Hawking and James Hartle, is that the boundary condition appropriate to our universe is that it should have no boundary. In other words, Euclidean space-time should be like the surface of a sphere (but with two extra dimensions to make it four-dimensional), -that is, finite in extent, but without a boundary or an edge. This is possible because in Euclidean space-time, space and time are on equal footing. The no-boundary proposal removes the need for choosing (and justifying) particular boundary conditions. It also suggests that, at least in imaginary time, the Big-Bang singularity is not a singularity at all. This is because a sphere – even a four-dimensional one – is a smooth object with no beginning; it has no sharp points or holes at which the laws of physics might break down. On the other hand, a singularity on the sphere would constitute a boundary to spacetime.
However, even if the no-boundary proposal is correct, to find the wavefunction of the universe we are still confronted with the problem of solving the Wheeler-De Witt equation, or equivalently, performing the sum over cosmic histories. Both involve considering an infinite number of degrees of freedom and are presently far beyond our capabilities. Instead, physicists consider only a finite subset of all these degrees of freedom, which, albeit unjustified, at least enables the equations to be solved. It is still too early to tell if this approach will be ultimately successful. The simple solutions which have been found are still too crude and unrealistic. They, nevertheless, provide much-needed insight into this fascinating field, whose frontiers lie well within the realm of yesteryear’s metaphysics.
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