Scale invariance in quantum field theory is broken by renormalization. A consequence is the irreversibility of the renormalization-group flow. There exists a function, counting the number of degrees of freedom of the theory, that decreases monotonically from the high to the low energies. In this paper I discuss this type of irreversibility and its origin in quantum field theory, in particular the so-called anomalies, which are quantum violations of classical conservation laws. The emerging irreversibility principle states that there is, in nature, a privileged spatial direction, pointing from the infinitely small to the arbitrarily large scales of magnitude.

1  Introduction

Quantum field theory defines a quantization procedure that begins with a classical, local lagrangian and ends with a non-local quantum functional. The formulation makes an essential use of the perturbative expansion, which, if applied naively, produces ``divergences'', when elementary vertices and quanta propagators are composed into complex interactions, the graphs. The theory has to be ``regularized'' by cutting out the frequencies greater than some 1/e, with e small, which are responsible for the infinities. Calculations have to be performed with a specific choice of the regularization procedure and a set of conventions for the removal of the divergences (``scheme''). It is possible to redefine the coupling constants so that they absorb all the infinities away, and the final results, when the cut-off e is removed in the limit e® 0, are finite and scheme-independent.

The coupling constants must satisfy a certain restriction, the renormalizability. The perturbative procedure for the removal of divergences generates a sort of feedback from the graphs to the vertices, which often requires that new vertices, and therefore new coupling constants, be added to the classical lagrangian. Only very few theories, the renormalizable ones, are stable with respect to this feedback and have a finite number of coupling constants.

The regularization of the theory requires that certain symmetries of the starting, ``classical'' lagrangian be relaxed. It can happen that when the cut-off is removed, some of the relaxed symmetries are no longer recoverable. This phenomenon is called anomaly and produces, in particular, the renormalization-group (RG) flow, the breakdown of scale invariance and the irreversibility. The purpose of this paper is to investigate these phenomena, which, I believe, might have a philosophical interest. The presentation of the notions of quantum field theory will necessarily be synthetic, and cannot replace the relevant literature. Useful works appeared in the philosophical literature, such as the papers of Teller (1988, 1989, 1990), Huggett and Weingard (1995) and Cao and Schweber (1993), which contain various mathematical issues (for example, see Teller, 1988, for divergent integrals, Huggett and Weingard, 1995, for the renormalization group and effective field theories), discussions of the physical and philosophical questions on renormalization, as well as a detailed historical survey (see in particular Cao and Schweber, 1993). My recent paper (Anselmi, 1999a) is also an introduction to the subject, but it has not been published, yet. Some common textbooks in quantum field theory are those of Bogoliubov and Shirkov (1976), Peskin and Schroeder (1995) and Collins (1984). It should be stressed, however, that familiarity with the techniques of quantum field theory is not necessary (and not even sufficient) to appreciate the conceptual structure of the arguments and, more importantly, their philosophical implications.

The irreversibility of the renormalization-group flow is a sort of spatial irreversibility. There exists a positive function, counting the degrees of freedom of the theory, which decreases monotonically with the energy. Anomalies, the renormalization-group flow and its irreversibility are presented in section 2. Section 3 contains a discussion about the intrinsic peculiarity of the RG flow and other observations on the new concept of irreversibility.

2  Anomalies and irreversibility

The renormalization-group flow is the dependence of the renormalized coupling constants and correlation functions on a reference scale m. The removal of divergences is obtained by performing a redefinition of the starting (``bare'') coupling constants of the theory. This redefinition of the couplings absorbs the infinities away and makes the correlation functions finite in the e® 0 limit. The new (``renormalized'') couplings are not ``constant'' anymore; actually, they can be interpreted as effective couplings depending on the energy scale. The RG equations are first-order equations and so the ``running'' couplings are completely specified once their values at some reference energy m are measured experimentally. In this way, a scale m enters the theory even if the starting ``classical'' lagrangian was scale-invariant.

Examples of classical scale-invariant theories are the theories of fields of spin 0, 1/2 and 1 whose interactions are schematically parametrized by a lagrangian of the form

L=  1 4 Fmn2+| ¶mj+ieAmj| 2+ y   ( ¶\slash +ieA\slash ) y+l1j4+l2j y   y.

(2.1)

No scale is present in L, because the couplings e,l₁,l₂ (which will be collectively denoted by l) have zero dimension in units of mass. This follows by assigning dimensions 1, 3/2 and 1 to the scalars j, fermions y and vectors A, respectively, while ¶_m º ¶/¶x^m has dimension 1, x^m has -1 and the action S=òd⁴x L is dimensionless.

The small- and large-distance limits of the theory are called ultraviolet (UV) and infrared (IR), respectively. They can be interchanged by a coordinate inversion, that is a transformation x^m® x^m/|x|², which implies the inversion of distances, |x|® 1/|x|, and exchanges, for example, the interior of the unit sphere |x|=1 with its exterior. The coordinate inversion is a symmetry of the theories (1). A world described by a theory (1) looks the same at the infinitesimal and arbitrarily large distances: we say that it is reversible in the spatial sense.

A renormalizable theory is allowed to have dimensionless coupling constants, plus parameters of positive dimension in units of mass (such as the masses themselves, for example). While a renormalizable theory of fields of spin 0, 1/2 and 1 such as (1) has no natural reference length l and is thus scale-invariant at the classical level, a mass m fixes a reference length l ~ 1/m and divides the energy range into two subdomains: at energies much greater than m, the effect of the mass is negligible; at energies much smaller than m, the massive fields are effectively suppressed (Appelquist and Carazzone, 1975): we can say that a mass kills degrees of freedom in the infrared, but does not modify the ultraviolet. A quick argument to show this is the following. Every configuration j(x) is weighted in quantum field theory by an exponential factor e^{[ i/((h/_2p) )]S[j]}. The action S[j] has to be finite, otherwise the contribution of the configuration j(x) is averaged away by the infinitely oscillating exponential factor. Now, the action of a free massive scalar field

ó õ d4x  æ è  1 2 (¶mj(x))2+  1 2 m2j(x)2 ö ø

(2.2)

stays finite in the IR limit (m® ¥) only if j is everywhere zero, which means that there is no j-propagation in this limit. On the other hand, the mass term is negligible in the UV limit (m® 0). Clearly, the lagrangian (2) is not invariant under the coordinate inversion x^m® x^m/|x|². The scale violations originated by dimensioned parameters will be called ``explicit'' violations.

An interpretation of these observations is that there are more degrees of freedom at high energies than at low energies, or that the number of degrees of freedom decreases with the energy. If we define a quantity a by counting the number of degrees of freedom as a function of the energy scale, it results that a is monotonically decreasing. In particular,

aUV ³ aIR.

(2.3)

Examples exhibiting the opposite behaviour are easy to find, however. A coupling constant with negative dimension in units of mass, such as Newton's constant in gravity, or the coupling of the j⁶-theory

ó õ d4x  æ è  1 2 (¶mj)2+  1 m2 j6 ö ø ,

acts in the opposite way: it kills degrees of freedom in the UV and leaves the IR unchanged. In this case, the function a will take larger values in the IR than in the UV: a_UV £ a_IR.

The intermediate situation is the one of a scale-invariant theory, such as (1). These theories do not exhibit, at the classical level, an interesting behaviour from the point of view of irreversibility and we always have a_UV=a_IR.

At the quantum level the results are of a different nature and the theories of class (1) become the most interesting ones, where a scale violation of an intrinsically different nature emerges. The quantization of a classically scale-invariant theory (1) does not produce, in general, a scale-invariant theory, because it is necessary to introduce a reference scale m at which the renormalized coupling constants are measured. The value of a coupling constant ``runs'', i.e. it depends on the energy. This dependence is given by the renormalization-group equations and in particular the beta function,

b(l(m))=m  dl(m) dm .

The quantum theory will no longer be invariant under the coordinate inversion x^m® x^m/|x|², although the classical lagrangian is. The question is to understand the effect of this breaking on the counter a of degrees of freedom. The function a will in general depend on l: a=a(l) and therefore on m. If a special behaviour such as (3) is obeyed by the theories (1), after quantization, then we can speak of ``irreversibility of the renormalization-group flow'', or, more briefly, quantum irreversibility.

It turns out, indeed, that a is monotonically decreasing along the RG flow, conventionally defined as the flow that points from the UV to the IR limit of the theory. The effect is qualitatively, but not quantitatively, similar to the effect of a mass. More precisely, the inequality (3) is obeyed, but the value of a_UV-a_IR is different from the value obtained by assimilating m to a mass. Particularly important to understanding the origin and meaning of this phenomenon are the so-called anomalies, which I discuss in detail.

Anomalies were discovered as violations of the axial symmetry. They are often called ``ABJ anomalies'', after Adler, Bell and Jackiw, who first realized their importance. Historical notes can be found in Jackiw (1985). Later, a similar violation of scale invariance was observed (Coleman and Jackiw, 1971) and, much later still, the closed formula () for this anomaly (called trace anomaly) was derived (Adler et al., 1977; Nielsen, 1977; Collins et al., 1977). The systematic study of anomalies in the context of quantum irreversibility is very recent (Anselmi, 1999b and 2000a).

A classical symmetry is associated with a conserved quantity Q(t), the charge, which a priori might depend on time. Conservation is the property that the time derivative of Q vanishes:

dQ dt =0.

(2.4)

Associated with a symmetry, there also exists a local operator, the current J_m(x), such that Q(t)=òd³x J₀(x). Conservation is exhibited by a stronger, local statement on the current, namely the vanishing of its divergence ¶_mJ^m=0.

Now, a classically vanishing quantity, such as the divergence of J^m or (4), need not be strictly zero in the regularized theory, where high frequencies (say, frequencies higher than some 1/e, whith e small) are cut away. This cutting, which allows us to classify the infinities as inverse powers of e, is in general very rough and breaks all sorts of symmetries. The most elegant regularization technique is the so-called dimensional continuation, largely developed by 't Hooft and Veltman (Collins, 1984). Nevertheless, no matter how elegant the chosen regularization technique is, it can be proved that it cannot preserve all the classical symmetries for e ¹ 0.

The conservation conditions broken by the regularization technique are violated in the regularized theory by arbitrarily small terms, proportional to e:

dQ dt =O(e),    or    ¶mJm=O(e).

(2.5)

Naively, the right-hand sides of (5) disappear when the cut-off is removed, by taking the limit e® 0. At first sight, it is reasonable to expect that the lost symmetries will be recovered in this limit. This is however false, but the matter is sufficiently subtle that some of the people who had studied the problem before Adler, Bell and Jackiw had not taken their own results seriously.

The divergences of the theory, which are arbitrarily large terms, such as 1/e, can conflict with the O(e) above, get simplified with those, and produce a finite non-vanishing result for [(dQ)/(dt)] and ¶_mJ^m, which will not disappear in the e® 0 limit. This violation of the conservation condition survives the entire quantization process and produces a new physical law of the theory, absent at the classical level.

Often, it is possible to show that the conservation condition (4) is identically true in the regularized theory. But this does not happen for all symmetries. The quantization implies a selection among the symmetries. The internal consistency of the theory, for example, demands that gauge symmetries be preserved, otherwise unitarity is lost. However, scale invariance does not belong to this list of ``protected'' symmetries and indeed it is violated, apart from very special cases.

Two peculiar cases where scale invariance is preserved at the quantum level are the infrared (large-distance) and ultraviolet (small-distance) limits of a running theory. In both cases the reference scale m is negligible, either because it is infinitely large or because it is infinitely small. Forcedly, the theory is scale-invariant in these limits. This means that the right-hand sides of (5) must be proportional to the beta function of the theory, which, by definition, vanishes in the IR and UV limits of the RG flow, and only there, where the renormalized coupling does not run: b(l_UV)=b(l_IR)=0. A typical beta function is shown in the figure. We can describe the renormalization-group flow as an interpolation between its two scale-invariant limits, pointing conventionally from the ultraviolet to the infrared ``conformal fixed points'' of the flow.

Observe that at the classical level all the theories (1) are scale-invariant, irrespectively of the values of their couplings l. At the quantum level, instead, only the theories with special values of the couplings, such as l_UV and l_IR, the zeros of the beta function, are scale-invariant. Conformal field theories can be free, weakly coupled, strongly interacting according to the values of l_UV and l_IR.

Formula (5) can effectively be replaced by expressions that do not contain the cut-off e and exhibit the surviving contribution in the e® 0 limit. Precisely,

¶mJm(x)=b(l) O(x),

where O(x) is some operator. We are interested here in the trace anomaly, where the current J^m is the stress tensor T_mn(x), which is classically conserved (¶_mT^mn=0) and traceless (Q º T_m^m=0) in the theories (1). The symmetries associated with these conservation conditions are the space-time translations and the scale transformations, respectively. In particular, the trace Q can be written as the divergence of the current generating the dilatations x^m® L x^m, of which the coordinate inversion x^m® x^m/|x|² is a particular case. As above, the right-hand sides of the conservation conditions can be of order e in the regularized theory (¶_mT^mn=O(e), Q = O(e)), and these ``evanescent'' quantities, when inserted into the correlation functions, can get multiplied by a divergence 1/e, thereby producing finite results. Now, it can be shown that translational invariance can always be preserved in the regularized theory, which means that the conservation condition ¶<sub>m</sub>T<sup>mn</sup>=0 does remain true at the quantum level. Instead, scale invariance is a less fundamental principle than translational invariance, to the extent that its violation cannot be avoided in running quantum field theories. This is, actually, the deep meaning of the ``running'', i.e. a non-trivial dependence on the scale m. The violation of the tracelessness condition reads, for the theories (1),

Q =  be 4 Fmn2-bl1j4-bl2j y   y,

(2.6)

where F_mn is the field strength and b_e,l1,l2 are the beta functions of the couplings e,l₁,l₂, respectively.

I remark again that the beta function is the coefficient of the terms in the trace anomaly. We can say that the beta function is the key-quantity to ``measure'' the effects of divergences and renormalization, the running of the coupling constant and the deviation from scale invariance. The critical points (UV and IR) are the points (b = 0) where scale-invariance is recovered and Q = 0. This condition is equivalent to saying that the coupling constant does not run at the fixed points and any dependence on the renormalization scale m disappears, as we have already remarked.

A deeper connection with the RG flow is established by the integrated trace anomaly òd⁴x Q(x), which is indeed equal to the m-derivative (Adler et al., 1977):

ó õ d4x Q(x)=m  ¶ ¶m .

This equation should be meant in the sense that the insertion of the operator òQ into a correlation function is equivalent to the m-derivative of the correlation function:

ó õ Q  f1¼fn =m  ¶ ¶m áf1¼fnñ.

In conclusion, the trace anomaly gives information about the dependence of the theory on the dynamical scale m. In particular, at criticality we have m¶/¶m® 0.

In view of these remarks, it is natural to expect that also the phenomenon of quantum irreversibility, which we are about to elucidate, will be essentially rooted in a property of the beta function. We will see, indeed, that the irreversible loss of degrees of freedom is measured by the area of the graph of the beta function between the fixed points (see the figure).

Figure

We are ready to introduce the counter a of degrees of freedom. It is convenient to couple the theory to external, non-quantized, fields. External fields are treated as sources for currents, such as J_m, T_mn or other composite operators, and are useful tools to simplify the operatorial equations. In our case, external fields are useful to give a simple definition of the function a. Our external field is gravity, since the metric tensor g_mn couples precisely to the stress tensor T_mn. Embedding the theory in external gravity is always possible, so that we are not losing generality by doing this.

The contribution in (6) is called internal anomaly, since the terms in this equation are dynamical operators. In the presence of external fields the trace anomaly contains other terms, similar to (6), constructed only with the external fields. They are not operators, but ordinary functions and their coefficients need not be proportional to the beta function.

The general form of the external contributions to the trace anomaly in the presence of a gravitational background involves two quantities, known as c and a, depending on the coupling constant l:

Q| ext=-  1 16p2 é ë c(l) Wmnrs2-a(l)  ~ R   2 mnrs  ù û .

(2.7)

The structure of the anomaly is uniquely fixed by general principles, but the values of c and a have to be calculated explicitly. W is the Weyl tensor and [R\tilde]_mnrs is the dual of the Riemann tensor. The term multiplied by a is known as the Euler density. To proceed with our discussion, it is not necessary to know more about these invariants. The UV and IR values of the functions c and a are given by c_UV=c(l_UV), a_UV=a(l_UV), etc.

The external contributions to the trace anomaly do not need to vanish at criticality, since external sources are passive spectators. Finite UV and IR values of c and a are perfectly compatible with the recovered scale invariance at the fixed points, since scale invariance requires only the vanishing of the internal contribution (6) to Q.

In a free-field theory the values of c and a are weighted, positive sums of the numbers of degrees of freedom. Precisely (Christensen and Duff, 1979):

c=  1 120 (ns+6nf+12nv),      a=  1 360 (ns+11nf+62nv),

(2.8)

where n_s, n_f and n_v are the numbers of real scalars, Dirac fermions and vectors, respectively.

In some cases the values of c and a have been calculated exactly in the IR limit of running theories (Anselmi et al., 1998a), which is in general a strongly interacting conformal field theory, and compared with the UV values given above. The UV limit is in this case a free-field theory (asymptotic freedom).

The simplest model is the supersymmetric version of QCD, identified by the number of colours N_c and quarks N_f: n_s = 4N_cN_f, n_f=N_cN_f+n_v/2 and n_v=N_c²-1 in the formulas (8). The conformal window is the range (3/2)N_c £ N_f £ 3N_c, where the beta function has two zeros, as shown in the figure: the free UV limit and the interacting IR theory. It is instructive to give a look at the total RG flows of c and a in this model, which are (Anselmi et al., 1998a):

cUV-cIR=-  NcNf 48 (1-3r)(9r2+3r-4),     aUV-aIR=  NcNf 48 (1-3r)2(2+3r),

(2.9)

where r=N_c/N_f. Without the simplifying properties of supersymmetry, the calculations of the total flows of c and a would require the resummation of infinitely many diagrams. A visible non-perturbative effect is the appearance of N_f in the denominator, absent in the free-field values (8) and at any finite order in perturbation theory.

The direct inspection of the formula (9) for a_UV-a_IR shows that the total flow of a is always positive, in the conformal-window range (3/2)N_c £ N_f £ 3N_c (it is not meaningful to use the formula outside of this range), while the total flow of c can have both signs in the conformal window. These conclusions were confirmed in all the theories where the strategy of the calculation could be applied (Anselmi et al., 1998b). At present, there are no other examples of exact results of this relevance, in particular in non-supersymmetric theories.

Let us now explain why the inequality (3), tested empirically in the models just mentioned, is a general property of the RG flow. A classical theory should have a positive-definite action, which ensures the existence of a minimum. The same property should be valid for the quantum functional G. Now, the regularization amounts to making integrals finite by cutting high frequencies away, or, which is the same, by compensating the contributions of high frequencies with opposite contributions. The subtractors cannot be physical particles; rather, they are negatively-normed states and therefore violate unitarity. The resulting regularized quantum action is not positive-definite, before renormalization. Renormalization can be seen as the restoration of the positive-definiteness of the quantum functional G. The precise statement is:

if the quantum functional G is positive-definite at some energy, then it is positive-definite at all energies.

Which means: if the coupling constants l are normalized to physical values (in general, they have to be positive, l > 0) at the reference energy scale m, G is positive-definite at arbitrary energies. This ensures that the quantization process produces a physically meaningful quantum theory starting from a physically meaningful classical theory.

Now, this property holds when all the fields are quantized. However, we are interested in the trace anomaly in the presence of external fields. Specifically we need the dependence of G on the external metric tensor. It is known that external fields give serious problems, to the extent that the above statement is generically false in this case.

This difficulty can be overcome in our case, because we do not actually need the complete dependence of G on the metric tensor g_mn, but just the dependence on its conformal factor f, which is precisely the external source coupled to Q. This means that we can put g_mn=e^2fd_mn, where d_mn is the flat-space metric. We denote the desired dependence by G[f]. The conformal factor f is very special, because it couples to an ``evanescent'' operator Q = O(e), that is an operator equal to zero in the classical, unregularized theory: the potential divergences are automatically cured by the evanescence of Q and the convergence of G[f] is not spoiled. Therefore, the quantum functional G[f] is positive-definite at all energies if it is positive-definite at some reference energy.

This implies a formula for the irreversibility of the RG flow, which reads (Anselmi, 1999b)

aUV-aIR=- ó õ lIR lUV  dl b(l)f(l)

(2.10)

where f is a metric in the space of coupling constants (in our case only one, l, but in general there could be many), defined by

áQ(x) Q(0)ñ =  1 15p4  b[l(t)]2 f[l(t)] |x|8

(2.11)

and t=ln|x|m. The denominator |x|⁸ is fixed by dimensionality (Q has dimension four), so that f is dimensionless. Positivity of the two-point function ensures that f > 0 throughout the RG flow, i.e. that f can indeed be viewed as a metric. Keeping in mind that in UV-free theories (l_UV=0, l_IR > 0) b is negative (see the figure), while in IR-free theories (l_UV > 0, l_IR=0) b is positive, the inequality (6) always follows.

Particularly appealing is the geometrical interpretation of formula (10): the a-defect is equal to the (positive) area of the graph of the beta function between the fixed points (see the figure). Recall that the removal of divergences depends on some arbitrary conventions, specifically the subtraction scheme. In particular, the beta function is not a physical quantity in itself, because it is scheme-dependent. Nevertheless, the integral (10) is scheme-independent, because of the presence of the metric f. It is possible to define a particularly nice scheme, where the metric f is constant throughout the RG flow, equal to its free-field (ultraviolet) value: f(l)=f(l_UV) º f_UV. This ``proper'' scheme is a special scheme resembling in some sense the ``proper'' time defined in General Relativity: it is the scheme in which the theory ``feels'' a constant metric f throughout the RG flow. The proper time in General Relativity is the time such that Ö{g<sub>00</sub>} (the ``metric'' for time intervals) is identically equal to unity throughout the life of a body. This observation shows that a renormalization scheme is very much like a coordinate frame and the scheme independence of the outcoming results is the statement that the physics cannot depend on the choice of coordinates.

Calling [`(l)], [`(b)], etc., the quantities calculated in the proper scheme, formula (10) reads

aUV-aIR=-fUV ó õ [`(l)]IR [`(l)]UV  d - l     - b   ( - l   ).

Apart from the factor in front of the integral, this expression is precisely the area of the graph of the ``proper'' beta function between the fixed points. Observe that the relative location of the extrema of integration is determined by the beta function itself ([`(l)]<sub>UV</sub> > [`(l)]<sub>IR</sub>Û b < 0 and [`(l)]<sub>UV</sub> < [`(l)]<sub>IR</sub>Û b > 0; see also the figure) and therefore the integral above is always positive and we can speak of ``area'' of the graph is a strict sense.

So, the net amount of massless degrees of freedom lost between two intermediate energies, corresponding to some values l₁ and l₂ of the running coupling constant, is the area of the graph contained between these values. A unit-area cell is associated with each massless degree of freedom lost along the RG flow. This is a quantization principle of a new type.

Formula (10) was checked to four loops (Anselmi, 1999b and 2000a), in four and six dimensions, in the most general renormalizable theory. In two dimensions an irreversibility theorem existed already (Zamolodchikov, 1986), which was called the c-theorem. However, this case was too simple to infer that a similar property was true in higher dimensions, and the question remained open (Cardy, 1988). With the exact results (9) of Anselmi et al. (1998a and b), the existence of a suitable higher-dimensional generalization became clear. Yet, those results did not prove inequality (6) and did not produce formula (10).

There is, indeed, a conceptually non-trivial jump, between the two and the four dimensions, connected with the fact that, roughly speaking, c and a are ``indistinguishable'' in two dimensions. Once the problem is correctly understood in four dimensions (Anselmi, 1998b), the ideas generalize straightforwardly to arbitrary even dimension (Anselmi, 2000a) and better explain the speciality of two dimensions (Anselmi, 2000b). There is a special subset of theories (those with c=a) in every even dimension, where most of the properties typical of two-dimensional conformal field theories hold; among these the fact that the RG flow can be assimilated to a mass flow, a fact that is in general not true.

Before discussing these issues, I conclude this section by mentioning that the results presented here might not generalize to three-dimensional quantum field theory, which is interesting for condensed matter applications. The reason is that formula (7) for the gravitational trace anomaly does not exist in odd dimensions, although the internal contribution has a similar form (for example, Q|_int=-b(l)j⁶ in the j⁶-theory, which is renormalizable in three dimensions), an analogue of (11) holds and, in principle, the final irreversibility formula (10) could also be written. It is an open problem to establish whether the RG flow is irreversible in three dimensions.

3  The intrinsic peculiarity of the renormalization-group flow

The quantity a can be interpreted as a counter of the degrees of freedom of the theory. The function a is positive and decreases from the small to the large distances. This suggests a new notion of irreversibility, an irreversibility in space, stating that there cannot exist a hypothetical reversed world, in which the small and large distances are interchanged with respect to our world.

There are several differences between quantum irreversibility and time irreversibility, the most important one being that quantum irreversibility if an exact law of nature.

Time irreversibility is very much intuitive, while quantum irreversibility does not appear to be intuitive, given the non-trivial technical work that we have done to identify the quantity a and establish its main properties. A quantum field theory expert is aware that renormalization introduces an asymmetry between the large and small distances, because divergences are only in the ultraviolet. In the Wilsonian approach to renormalization, moreover, the RG flow is defined by integrating out degrees of freedom, up to some energy scale and this argument is sometimes advocated in favour of the claimed irreversibility (Peskin and Schroder, 1995). I think, however, that this argument is not sufficient to suggest that the RG flow is irreversibile, or at least it does not suggest it as neatly as the observation of the world suggests that time is irreversible. First, the intrinsic irreversibility of the integrating-out procedure is not so clear and certainly disputable. Second, the integrated-out degrees of freedom have not ``disappeared'': they are still around and their contribution is not quantified to be less than before the integration. Third, no appropriate counter of the degrees of freedom is outlined in this approach. This is not a minor point, because the quantity c, for example, is always positive, it could be a good counter of the degrees of freedom, it does decrease along the massive flows (2) (which are the clearest analogue of the Wilsonian integrating-out procedure), but it does not decrease along RG flows (see (9)). This means that there must be something subtle here and if an intuitive argument in favour of a principle of spatial irreversibility exists, it should be much simpler and follow from the general principles of quantum mechanics and the observation of the world, leaving the technical aspects of the approaches to quantum field theory aside.

We have already stressed that there is no place, in a classical context, for the new idea of irreversibility. Nevertheless, this idea becomes intuitive at the quantum level, in the following sense. We know that the effects of the indeterminacy principle are depressed and averaged away at the large scales of magnitude, where determinism prevails. This can be better appreciated when comparing a single indeterministic system (atom) to an ensemble made of a large number of indeterministic systems, which appears in general to be deterministic. Certainly, indeterministic phenomena can be visible macroscopically (this happens, for example, in every experiment of quantum mechanics), but this event is statistically disfavoured. The world becomes less and less ``free'' when we consider larger and larger scales of magnitude and this is a clear indication in favour of a spatial irreversibility.

Scale invariance can be associated, by analogy, to the notion of ``reversibility'', since it asserts that the infinitesimally small and arbitrarily large distances are interchangeable, by the coordinate inversion x<sup>m</sup>® x<sup>m</sup>/|x|<sup>2</sup>, and that this change is physically unobservable. We have just remarked that the violation of this interchangeability is an implicit consequence of the indeterminacy principle, in complex systems of a large number of atoms. Nevertheless, we know that quantum mechanics is not logically closed and does not describe systems of infinitely many quanta, while in quantum field theory there is no restriction on the number of quanta. Therefore, it is natural to expect that a rigorous formulation of the ``intuitive'' idea of quantum irreversibility can only come from quantum field theory and it is not surprising that this requires a very non-trivial technical work.

Concluding, space, or quantum, irreversibility, inconceivable at the classical level, becomes intuitive at the quantum level and rigorously established in quantum field theory. The phenomena of high-energy physics are deeply different from those of the large scales of magnitude and there exists a universal principle dictating this change. This principle does not change with the scales of magnitude. More concretely, the value of the quantity a changes with distances, but the way in which it changes, i.e. the sign of its derivative, does not change.

Observe that the process leading to irreversible laws (the quantum functional G) starting from reversible ones (the classical lagrangian L) is, in quantum field theory, purely conceptual, or mathematical: it is the very same quantization process and perturbative definition of the theory. Strictly speaking, the intermediate steps can have no direct physical meaning, to the extent that the starting classical lagrangian (e.g. the theory of quarks and gluons in QCD) might not be related to any observed physical phenomenon. Only the final quantum action G carries information about the exact physical laws. That is why, in particular, quantum irreversibility is an exact law of nature, while classical reversibility is approximate. In statistical thermodynamics, indeed, the roles are inverted: the starting, microscopic laws, are the true, exact laws of nature, and so the outcoming irreversible laws are necessarily approximate. Time irreversibility is not an exact law of nature, but quantum irreversibility is. Anomalies and quantum irreversibility are a remarkable illustration of how divergences can ``change the world'', to the extent that they make it spatially irreversible.

Effects of explicit violations might superpose to the genuine effect of anomalies on irreversibility. This is, for example, the case of the mass terms, which enhance inequality (6), while other dimensionful parameters (Newton's constant) can violate it. Explicit effects, however, do not obey formula (10), which is specific of the RG flow. For this reason the dynamical scale m is intrinsically different from any dimensionful parameter. The difference between the irreversibility induced by the dynamical scale m and the effects of explicit scales can be rigorously proved (Anselmi, 2000b). It happens that the effect of a mass, for example, on the difference a_UV-a_IR is quantitatively different from the effect of the RG scale m.

On the other hand, it can be shown that the two effects are exactly equal in the theories that have c=a. They are always equal in two dimensions. This is an important remark, because it shows that the two-dimensional c-theorem (Zamolodchikov, 1986) does not reveal the intrinsic peculiarity of the irreversibility of the RG flow. Only in four dimensions, and for c ¹ a , can we appreciate this deep peculiarity.

It is easy to see that the equality of c and a is incompatible with the known experimental results. It is sufficient to apply the free-field formulas (8) at the various energies that have been explored so far. The equality of c and a would require, in particular, that when some fermionic particle becomes relevant to the counting of degrees of freedom (for example, the electron becomes relevant at energies greater than its mass), some new bosonic field should appear at the same time, in order to maintain a certain balance between bosons and fermions, as implied by c=a in (8). This sort of balance between bosons and fermions, however, is in contradiction with the experimental observations.

If the world had c=a, there would be nothing special in the scale m and it could be assimilated to a mass. We would be allowed to say that divergences are really a blunder and can be traded for a purely classical effect, maybe formulating the theory by means of different variables. Then, the efforts done by several physicists, typically string theorists, to prove that ``quantum = classical in disguise'' would have some meaning. Unfortunately, or fortunately, depending on taste, nature shows that c and a are different. Now, given that string theory is divergence-free, it is also clear that it misses the peculiarity of the RG flow and can only simulate the effect of a missing dynamical scale m with explicit scales. The statement that this simulation cannot be quantitatively precise at c ¹ a, in spite of being qualitatively acceptable, casts severe doubt on the real power of string theory to describe even a part of nature in an exact way. Rather, in the light of our observations, string theory better sounds like a very limited and approximative theoretical set-up for low-energy phenomena.

The results showing the speciality of the RG flow might have, in the context of renormalization, the same relevance as the Bell inequalities in quantum mechanics. These inequalities proved that quantum indeterminacy cannot be assimilated to a pseudo-classical theory of (local) hidden variables. It was therefore possible to discriminate between quantum indeterminacy and local hidden variables experimentally. Similarly, the results of Anselmi (2000b) show that the renormalization-group flow cannot be assimilated to a massive flow, if c ¹ a.

We have discussed, so far, the dimensionless, i.e. strictly renormalizable, parameters, which single out the specificity of the RG flow, and the super-renormalizable parameters, such as the masses. Let us finally comment on those parameters that have negative dimensions in units of mass, such as the Newton constant. We have remarked that these parameters might violate the inequality (3). If this happened for gravity, for example, then we would not be allowed to say that quantum irreversibility is a principle of nature, although it would still remain a completely general property of the RG flow. However, although we know very little about quantum gravity, there is a natural way in which gravity might be included in the treatment, which I now explain, so that it is reasonable to think that we have outlined a new principle of nature.

Gravity will obey the irreversibility property if the Newton constant is not a fundamental constant, but only a low-energy effect of the dynamical RG scale m. This idea is suggested by an analogy with massless QCD, i.e. the theory of quantum chromodynamics with massless quarks. The hadrons do have masses, at low energies, but these are all proportional to the dynamical scale of the theory, generated by m, commonly denoted by L _QCD: masses can be generated (``simulated'') by m, but not vice versa. A low-energy theory of pions looks necessarily non-renormalizable, but its non-renormalizable parameters are not fundamental constants of nature; rather they are generated by m in the quantization of a classically scale-invariant (i.e. ``reversible'') high-energy theory. It is tempting to guess that something similar happens from gravity. It is therefore conceivable that no non-renormalizable parameter is a fundamental constant of nature or, going further, that no dimensioned parameter is fundamental (masses included) and all scales of nature descend from m.

We can rephrase this statement as a restriction on the correspondence principle: all the fundamental theories of nature should be obtained by quantizing scale-invariant classical theories, so that all the dimensioned parameters descend from the unique, dynamical scale m. It is remarkable that there already exists an example of ``perfect'' theory from this point of view, namely massless QCD. The ultimate theory of the world might be precisely like massless QCD.

QCD is formulated from the ultraviolet, in the sense that the perturbative expansion is an expansion around the UV limit, which is free. All other theories, in particular QED, the Standard Model and the j⁴-theory, are IR-free and formulated as perturbative expansions around their low-energy limits. This, however, conflicts with quantum irreversibility, because some degrees of freedom are lost from high to low energies. It is natural to expect that the theories formulated from the IR should exhibit, sooner or later, problems related to the missing degrees of freedom, such as singularities of some type, violations of unitarity, etc. It seems that in QED such problems are the presence of ghosts in the theory (the Landau pole), while the j⁴-theory might be non-perturbatively trivial. The problem of gravity, on the other hand, is its non-renormalizability, but in some sense the irreversibility of the RG flow shows that this problem is not more serious than the problems of QED and the Standard Model, because only QCD is completely consistent with the present knowledge. The message of quantum irreversibility is that a consistent formulation of quantum field theory should necessarily start from the ultraviolet and descend towards the infrared.

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