Black Hole Entropy in terms of Mass. Note Quote.

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If M-theory is compactified on a d-torus it becomes a D = 11 – d dimensional theory with Newton constant

GD = G11/Ld = l911/Ld —– (1)

A Schwartzschild black hole of mass M has a radius

Rs ~ M(1/(D-3)) GD(1/(D-3)) —– (2)

According to Bekenstein and Hawking the entropy of such a black hole is

S = Area/4GD —– (3)

where Area refers to the D – 2 dimensional hypervolume of the horizon:

Area ~ RsD-2 —– (4)

Thus

S ~ 1/GD (MGD)(D-2)/(D-3) ~ M(D-2)/(D-3) GD1/(D-3) —– (5)

From the traditional relativists’ point of view, black holes are extremely mysterious objects. They are described by unique classical solutions of Einstein’s equations. All perturbations quickly die away leaving a featureless “bald” black hole with ”no hair”. On the other hand Bekenstein and Hawking have given persuasive arguments that black holes possess thermodynamic entropy and temperature which point to the existence of a hidden microstructure. In particular, entropy generally represents the counting of hidden microstates which are invisible in a coarse grained description. An ultimate exact treatment of objects in matrix theory requires a passage to the infinite N limit. Unfortunately this limit is extremely difficult. For the study of Schwarzchild black holes, the optimal value of N (the value which is large enough to obtain an adequate description without involving many redundant variables) is of order the entropy, S, of the black hole.

Considering the minimum such value for N, we have

Nmin(S) = MRs = M(MGD)1/D-3 = S —– (6)

We see that the value of Nmin in every dimension is proportional to the entropy of the black hole. The thermodynamic properties of super Yang Mills theory can be estimated by standard arguments only if S ≤ N. Thus we are caught between conflicting requirements. For N >> S we don’t have tools to compute. For N ~ S the black hole will not fit into the compact geometry. Therefore we are forced to study the black hole using N = Nmin = S.

Matrix theory compactified on a d-torus is described by d + 1 super Yang Mills theory with 16 real supercharges. For d = 3 we are dealing with a very well known and special quantum field theory. In the standard 3+1 dimensional terminology it is U(N) Yang Mills theory with 4 supersymmetries and with all fields in the adjoint repersentation. This theory is very special in that, in addition to having electric/magnetic duality, it enjoys another property which makes it especially easy to analyze, namely it is exactly scale invariant.

Let us begin by considering it in the thermodynamic limit. The theory is characterized by a “moduli” space defined by the expectation values of the scalar fields φ. Since the φ also represents the positions of the original DO-branes in the non compact directions, we choose them at the origin. This represents the fact that we are considering a single compact object – the black hole- and not several disconnected pieces.

The equation of state of the system, defined by giving the entropy S as a function of temperature. Since entropy is extensive, it is proportional to the volume ∑3 of the dual torus. Furthermore, the scale invariance insures that S has the form

S = constant T33 —– (7)

The constant in this equation counts the number of degrees of freedom. For vanishing coupling constant, the theory is described by free quanta in the adjoint of U(N). This means that the number of degrees of freedom is ~ N2.

From the standard thermodynamic relation,

dE = TdS —– (8)

and the energy of the system is

E ~ N2T43 —– (9)

In order to relate entropy and mass of the black hole, let us eliminate temperature from (7) and (9).

S = N23((E/N23))3/4 —– (10)

Now the energy of the quantum field theory is identified with the light cone energy of the system of DO-branes forming the black hole. That is

E ≈ M2/N R —– (11)

Plugging (11) into (10)

S = N23(M2R/N23)3/4 —– (12)

This makes sense only when N << S, as when N >> S computing the equation of state is slightly trickier. At N ~ S, this is precisely the correct form for the black hole entropy in terms of the mass.

F-Theory Compactifications on Calabi-Yau Manifolds Capture Nonperturbative Physics of String Theory. Note Quote.

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The distinct string theory and their strong-coupling limit. The solid line (-) denotes toroidal compactification, the dashed line (–) denotes K3 compactifications and the dotted line (…) denotes Y3 compactifications. The fine-dotted line (…) denotes Y4 compactifications while the horizontal bar (-) indicates a string-string duality. The theories marked with a ‘U’ (‘8’) have a U-duality (8-duality); the strong-coupling limit of the theories marked by ‘M’ (‘F’) are controlled by M-theory (F-theory).

The type-IIB theory in 10 spacetime dimensions is believed to have an exact SL(2, Z) quantum symmetry which acts on the complex scalar τ = e-2φ + iφ’, where φ and φ’ are the two scalar fields of type-lIB theory. This fact led Vafa to propose that the type-lIB string could be viewed as the toroidal compactification of a twelve-dimensional theory, called F-theory, where T is the complex structure modulus of a two-torus T2 and the Kähler-class modulus is frozen. Apart from having a geometrical interpretation of the SL(2, Z) symmetry this proposal led to the construction of new, nonperturbative string vacua in lower space-time dimensions. In order to preserve the SL(2, Z) quantum symmetry the compactification manifold cannot be arbitrary but has to be what is called an elliptic fibration. That is, the manifold is locally a fibre bundle with a two-torus T2 over some base B but there are a finite number of singular points where the torus degenerates. As a consequence nontrivial closed loops on B can induce a SL(2, Z) transformation of the fibre. This implies that the dilaton is not constant on the compactification manifold, but can have SL(2, Z) monodromy. It is precisely this fact which results in nontrivial (nonperturbative) string vacua inaccessible in string perturbation theory.

F-theory can be compactified on elliptic Calabi-Yau manifolds and each of such compactifications is conjectured to capture the nonperturbative physics of an appropriate string vacua. One finds:

10 or 11 Dimensions? Phenomenological Conundrum. Drunken Risibility.

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It is not the fact that we are living in a ten-dimensional world which forces string theory to a ten-dimensional description. It is that perturbative string theories are only anomaly-free in ten dimensions; and they contain gravitons only in a ten-dimensional formulation. The resulting question, how the four-dimensional spacetime of phenomenology comes off from ten-dimensional perturbative string theories (or its eleven-dimensional non-perturbative extension: the mysterious M theory), led to the compactification idea and to the braneworld scenarios.

It is not the fact that empirical indications for supersymmetry were found, that forces consistent string theories to include supersymmetry. Without supersymmetry, string theory has no fermions and no chirality, but there are tachyons which make the vacuum instable; and supersymmetry has certain conceptual advantages: it leads very probably to the finiteness of the perturbation series, thereby avoiding the problem of non-renormalizability which haunted all former attempts at a quantization of gravity; and there is a close relation between supersymmetry and Poincaré invariance which seems reasonable for quantum gravity. But it is clear that not all conceptual advantages are necessarily part of nature – as the example of the elegant, but unsuccessful Grand Unified Theories demonstrates.

Apart from its ten (or eleven) dimensions and the inclusion of supersymmetry – both have more or less the character of only conceptually, but not empirically motivated ad-hoc assumptions – string theory consists of a rather careful adaptation of the mathematical and model-theoretical apparatus of perturbative quantum field theory to the quantized, one-dimensionally extended, oscillating string (and, finally, of a minimal extension of its methods into the non-perturbative regime for which the declarations of intent exceed by far the conceptual successes). Without any empirical data transcending the context of our established theories, there remains for string theory only the minimal conceptual integration of basic parts of the phenomenology already reproduced by these established theories. And a significant component of this phenomenology, namely the phenomenology of gravitation, was already used up in the selection of string theory as an interesting approach to quantum gravity. Only, because string theory – containing gravitons as string states – reproduces in a certain way the phenomenology of gravitation, it is taken seriously.

But consistency requirements, the minimal inclusion of basic phenomenological constraints, and the careful extension of the model-theoretical basis of quantum field theory are not sufficient to establish an adequate theory of quantum gravity. Shouldn’t the landscape scenario of string theory be understood as a clear indication, not only of fundamental problems with the reproduction of the gauge invariances of the standard model of quantum field theory (and the corresponding phenomenology), but of much more severe conceptual problems? Almost all attempts at a solution of the immanent and transcendental problems of string theory seem to end in the ambiguity and contingency of the multitude of scenarios of the string landscape. That no physically motivated basic principle is known for string theory and its model-theoretical procedures might be seen as a problem which possibly could be overcome in future developments. But, what about the use of a static background spacetime in string theory which falls short of the fundamental insights of general relativity and which therefore seems to be completely unacceptable for a theory of quantum gravity?

At least since the change of context (and strategy) from hadron physics to quantum gravity, the development of string theory was dominated by immanent problems which led with their attempted solutions deeper. The result of this successively increasing self- referentiality is a more and more enhanced decoupling from phenomenological boundary conditions and necessities. The contact with the empirical does not increase, but gets weaker and weaker. The result of this process is a labyrinthic mathematical structure with a completely unclear physical relevance

Duality’s Anti-Realism or Poisoning Ontological Realism: The Case of Vanishing Ontology. Note Quote.

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If the intuitive quality of the external ontological object is diminished piece by piece during the evolutionary progress of physical theory (which must be acknowledged also in a hidden parameter framework), is there any core of the notion of an ontological object at all that can be trusted to be immune against scientific decomposition?

Quantum mechanics cannot answer this question. Contemporary physics is in a quite different position. The full dissolution of ontology is a characteristic process of particle physics whose unfolding starts with quantum mechanics and gains momentum in gauge field theory until, in string theory, the ontological object has simply vanished.

The concept to be considered is string duality, with the remarkable phenomenon of T-duality according to which a string wrapped around a small compact dimension can as well be understood as a string that is not wrapped but moves freely along a large compact dimension. The phenomenon is rooted in the quantum principles but clearly transcends what one is used to in the quantum world. It is not a mere case of quantum indeterminacy concerning two states of the system. We rather face two theoretical formulations which are undistinguishable in principle so that they cannot be interpreted as referring to two different states at all. Nevertheless the two formulations differ in characteristics which lie at the core of any meaningful ontology of an external world. They differ in the shape of space-time and they differ in form and topological position of the elementary objects. The fact that those characteristics are reduced to technical parameters whose values depend on the choice of the theoretical formulation contradicts ontological scientific realism in the most straightforward way. If a situation can be described by two different sets of elementary objects depending on the choice of the theoretical framework, how can it make sense to assert that these ontological objects actually exist in an external world?

The question gets even more virulent as T-duality by no means remains the only duality relation that surfaces in string theory. It turns out that the existence of dualities is one of string theory’s most characteristic features. They seem to pop up wherever one looks for them. Probably the most important role played by duality relations today is to connect all different superstring theories. Before 1995 physicists knew 5 different types of superstring theory. Then it turned out that these 5 theories and a 6th by then unknown theory named ‘M-theory’ are interconnected by duality relations. Two types of duality are involved. Some theories can be transformed into each other through inversion of a compactification radius, which is the phenomenon we know already under the name of T-duality. Others can be transformed into each other by inversion of the string coupling constant. This duality is called S-duality. Then there is M-theory, where the string coupling constant is transformed into an additional 11th dimension whose size is proportional to the coupling strength of the dual theory. The described web of dualities connects theories whose elementary objects have different symmetry structure and different dimensionality. M-theory even has a different number of spatial dimensions than its co-theories. Duality nevertheless implies that M-theory and the 5 possible superstring theories only represent different formulations of one single actual theory. This statement constitutes the basis for string theory’s uniqueness claims and shows the pivotal role played by the duality principle.

An evaluation of the philosophical implications of duality in modern string theory must first acknowledge that the problems to identify uniquely the ontological basis of a scientific theory are as old as the concept of invisible scientific objects itself. Complex theories tend to allow the insertion of ontology at more than one level of their structure. It is not a priori clear in classical electromagnetism whether the field or the potential should be understood as the fundamental physical object and one may wonder similarly in quantum field theory whether that concept’s basic object is the particle or the field. Questions of this type clearly pose a serious philosophical problem. Some philosophers like Quine have drawn the conclusion to deny any objective basis for the imputation of ontologies. Philosophers with a stronger affinity for realism however often stress that there do exist arguments which are able to select a preferable ontological set after all. It might also be suggested that ontological alternatives at different levels of the theoretical structure do not pose a threat to realism but should be interpreted merely as different parameterisations of ontological reality. The problem is created at a philosophical level by imputing an ontology to a physical theory whose structure neither depends on nor predetermines uniquely that imputation. The physicist puts one compact theoretical structure into space-time and the philosopher struggles with the question at which level ontological claims should be inserted.

The implications of string-duality have an entirely different quality. String duality really posits different ‘parallel’ empirically indistinguishable versions of structure in spacetime which are based on different sets of elementary objects. This statement is placed at the physical level independently of any philosophical interpretation. Thus it transfers the problem of the lack of ontological uniqueness from a philosophical to a physical level and makes it much more difficult to cure. If theories with different sets of elementary objects give the same physical world (i. e. show the same pattern of observables), the elementary object cannot be seen as the unique foundation of the physical world any more. There seems to be no way to avoid this conclusion. There exists an additional aspect of duality that underlines its anti-ontological character. Duality does not just spell destruction for the notion of the ontological scientific object but in a sense offers a replacement as well.

Do there remain any loop-holes in duality’s anti-realist implications which could be used by the die-hard realist? A natural objection to the asserted crucial philosophical importance of duality can be based on the fact, that duality was not invented in the context of string theory. It is known since the times of P. M. Dirac that quantum electrodynamics with magnetic monopoles would be dual to a theory with inverted coupling constant and exchanged electric and magnetic charges. The question arises, if duality is poison to ontological realism, why didn’t it have its effect already at the level of quantum electrodynamics. The answer gives a nice survey of possible measures to save ontological realism. As it will turn out, they all fail in string theory.

In the case of quantum-electrodynamics the realist has several arguments to counter the duality threat. First, duality looks more like an accidental oddity that appears in an unrealistic scenario than like a characteristic feature of the world. No one has observed magnetic monopoles, which renders the problem hypothetical. And even if there were magnetic monopoles, an embedding of electromagnetism into a fuller description of the natural forces would destroy the dual structure anyway.

In string theory the situation is very different. Duality is no ‘lucky strike’ any more, which just by chance arises in a certain scenario that is not the real one anyway. As we have seen, it rather represents a core feature of the emerging theoretical structure and cannot be ignored. A second option open to the realist at the level of quantum electrodynamics is to shift the ontological posit. Some philosophers of quantum physics argue that the natural elementary object of quantum field theory is the quantum field, which represents something like the potentiality to produce elementary particles. One quantum field covers the full sum over all variations of particle exchange which have to be accounted for in a quantum process. The philosopher who posits the quantum field to be the fundamental real object discovered by quantum field theory understands the single elementary particles as mere mathematical entities introduced to calculate the behaviour of the quantum field. Dual theories from his perspective can be taken as different technical procedures to calculate the behaviour of the univocal ontological object, the electromagnetic quantum field. The phenomenon of duality then does not appear as a threat to the ontological concept per se but merely as an indication in favour of an ontologisation of the field instead of the particle.

The field theoretical approach to interpret the quantum field as the ontological object does not have any pendent in string theory. String theory only exists as a perturbative theory. There seems to be no way to introduce anything like a quantum field that would cover the full expansion of string exchanges. In the light of duality this lack of a unique ontological object arguably appears rather natural. The reason is related to another point that makes string dualities more dramatic than its field theoretical predecessor. String theory includes gravitation. Therefore object (the string geometry) and space-time are not independent. Actually it turns out that the string geometry in a way carries all information about space-time as well. This dependence of space-time on string-geometry makes it difficult already to imagine how it should be possible to put into this very spacetime some kind of overall field whose coverage of all string realisations actually implies coverage of variations of spacetime itself. The duality context makes the paradoxical quality of such an attempt more transparent. If two dual theories with different radii of a compactified dimension shall be covered by the same ontological object in analogy to the quantum field in field theory, this object obviously cannot live in space and time. If it would, it had to choose one of the two spacetime versions endorsed by the dual theories, thereby discriminating the other one. This theory however should not be expected to be a theory of objects in spacetime and therefore does not rise any hopes to redeem the external ontological perspective.

A third strategy to save ontological realism is based on the following argument: In quantum electrodynamics the difference between the dual theories boils down to a mere replacement of a weak coupling constant which allows perturbative calculation by a strong one which does not. Therefore the choice is open between a natural formulation and a clumsy untreatable one which maybe should just be discarded as an artificial construction.

Today string theory cannot tell whether its final solution will put its parameters comfortably into the low-coupling-constant-and-large-compact-dimension-regime of one of the 5 superstring theories or M-theory. This might be the case but it might as well happen, that the solution lies in a region of parameter space where no theory clearly stands out in this sense. However, even if there was one preferred theory, the simple discarding of the others could not save realism as in the case of field theory. First, the argument of natural choice is not really applicable to T-duality. A small compactification radius does not render a theory intractable like a large coupling constant. The choice of the dual version with a large radius thus looks more like a convention than anything else. Second, the choice of both compactification radii and string coupling constants in string theory is the consequence of a dynamical process that has to be calculated itself. Calculation thus stands before the selection of a certain point in parameter space and consequently also before a possible selection of the ontological objects. The ontological objects therefore, even if one wanted to hang on to their meaningfulness in the final scenario, would appear as a mere product of prior dynamics and not as a priori actors in the game.

Summing up, the phenomenon of duality is admittedly a bit irritating for the ontological realist in field theory but he can live with it. In string theory however, the field theoretical strategies to save realism all fail. The position assumed by the duality principle in string theory clearly renders obsolete the traditional realist understanding of scientific objects as smaller cousins of visible ones. The theoretical posits of string theory get their meaning only relative to their theoretical framework and must be understood as mathematical concepts without any claim to ‘corporal’ existence in an external world. The world of string theory has cut all ties with classical theories about physical bodies. To stick to ontological realism in this altered context, would be inadequate to the elementary changes which characterize the new situation. The demise of ontology in string theory opens new perspectives on the positions where the stress is on the discontinuity of ontological claims throughout the history of scientific theories.