Mappings, Manifolds and Kantian Abstract Properties of Synthesis

An inverse system is a collection of sets which are connected by mappings. We start off with the definitions before relating these to abstract properties of synthesis.

Definition: A directed set is a set T together with an ordering relation ≤ such that

(1) ≤ is a partial order, i.e. transitive, reflexive, anti-symmetric

(2) ≤ is directed, i.e. for any s, t ∈ T there is r ∈ T with s, t ≤ r

Definition: An inverse system indexed by T is a set D = {D_s|s ∈ T} together with a family of mappings F = {h_st|s ≥ t, h_st : D_s → D_t}. The mappings in F must satisfy the coherence requirement that if s ≥ t ≥ r, h_tr ◦ h_st = h_sr.

Interpretation of the index set: The index set represents some abstract properties of synthesis. The ‘synthesis of apprehension in intuition’ proceeds by a ’running through and holding together of the manifold’ and is thus a process that takes place in time. We may now think of an index s ∈ T as an interval of time available for the process of ’running through and holding together’. More formally, s can be taken to be a set of instants or events, ordered by a ‘precedes’ relation; the relation t ≤ s then stands for: t is a substructure of s. It is immediate that on this interpretation ≤ is a partial order. The directedness is related to what Kant called ‘the formal unity of the consciousness in the synthesis of the manifold of representations’ or ‘the necessary unity of self-consciousness, thus also of the synthesis of the manifold, through a common function of the mind for combining it in one representation’ – the requirement that ‘for any s, t ∈ T there is r ∈ T with s, t ≤ r’ creates the formal conditions for combining the syntheses executed during s and t in one representation, coded by r.

Interpretation of the D_s and the mappings h_st : D_s → D_t. An object in D_s can thought of as a possible ‘indeterminate object of empirical intuition’ synthesised in the interval s. If s ≥ t, the mapping h_st : D_s → D_t expresses a consistency requirement: if d ∈ D_s represents an indeterminate object of empirical intuition synthesised in interval s, so that a particular manifold of features can be ‘run through and held together’ during s, some indeterminate object of empirical intuition must already be synthesisable by ‘running through and holding together’ in interval t, e.g. by combining a subset of the features characaterising d. This interpretation justifies the coherence condition s ≥ t ≥ r, h_tr ◦ h_st = h_sr: the synthesis obtained from first restricting the interval available for ‘running through and holding together’ to interval t, and then to interval r should not differ from the synthesis obtained by restricting to r directly.

We do not put any further requirements on the mappings h_st : D_s → D_t, such as surjectivity or injectivity. Some indeterminate object of experience in D_t may have disappeared in D_s: more time for ‘running through and holding together’ may actually yield fewer features that can be combined. Thus we do not require the mappings to be surjective. It may also happen that an indeterminate object of experience in D_t corresponds to two or more of such objects in D_s, as when a building viewed from afar upon closer inspection turns out to be composed of two spatially separated buildings; thus the mappings need not be injective.

The interaction of the directedness of the index set and the mappings h_st is of some interest. If r ≥ s, t there are mappings h_rs : D_r → D_s and h_rt : D_s → D_t. Each ‘indeterminate object of empirical intuition’ in d ∈ D_r can be seen as a synthesis of such objects h_rs(d) ∈ D_s and h_rt(d) ∈ D_t. For example, the ‘manifold of a house’ can be viewed as synthesised from a ‘manifold of the front’ and a ‘manifold of the back’. The operation just described has some of the characteristics of the synthesis of reproduction in imagination: the fact that the front of the house can be unified with the back to produce a coherent object presupposes that the front can be reproduced as it is while we are staring at the back. The mappings h_rs : D_r → D_s and h_rt : D_s → D_t capture the idea that d ∈ D_r arises from reproductions of h_rs(d) and h_rt(d) in r.

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Expressivity of Bodies: The Synesthetic Affinity Between Deleuze and Merleau-Ponty. Thought of the Day 54.0

It is in the description of the synesthetic experience that Deleuze finds resources for his own theory of sensation. And it is in this context that Deleuze and Merleau-Ponty are closest. For Deleuze sees each sensation as a dynamic evolution, sensation is that which passes from one ‘order’ to another, from one ‘level’ to another. This means that each sensation is at diverse levels, of different orders, or in several domains….it is characteristic of sensation to encompass a constitutive difference of level and a plurality of constituting domains. What this means for Deleuze is that sensations cannot be isolated in a particular field of sense; these fields interpenetrate, so that sensation jumps from one domain to another, becoming-color in the visual field or becoming-music on the auditory level. For Deleuze (and this goes beyond what Merleau-Ponty explicitly says), sensation can flow from one field to another, because it belongs to a vital rhythm which subtends these fields, or more precisely, which gives rise to the different fields of sense as it contracts and expands, as it moves between different levels of tension and dilation.

If, as Merleau-Ponty says (and Deleuze concurs), synesthetic perception is the rule, then the act of recognition that identifies each sensation with a determinate quality or sense and operates their synthesis within the unity of an object, hides from us the complexity of perception, and the heterogeneity of the perceiving body. Synesthesia shows that the unity of the body is constituted in the transversal communication of the senses. But these senses are not pre given in the body; they correspond to sensations that move between levels of bodily energy – finding different expression in each other. To each of these levels corresponds a particular way of living space and time; hence the simultaneity in depth that is experienced in vision is not the lateral coexistence of touch, and the continuous, sensuous and overlapping extension of touch is lost in the expansion of vision. This heterogenous multiplicity of levels, or senses, is open to communication; each expresses its embodiment in its own way, and each expresses differently the contents of the other senses.

Thus sensation is not the causal process, but the communication and synchronization of senses within my body, and of my body with the sensible world; it is, as Merleau-Ponty says, a communion. And despite frequent appeal in the Phenomenology of Perception to the sameness of the body and to the common world to ground the diversity of experience, the appeal here goes in a different direction. It is the differences of rhythm and of becoming, which characterize the sensible world, that open it up to my experience. For the expressive body is itself such a rhythm, capable of synchronizing and coexisting with the others. And Merleau-Ponty refers to this relationship between the body and the world as one of sympathy. He is close here to identifying the lived body with the temporization of existence, with a particular rhythm of duration; and he is close to perceiving the world as the coexistence of such temporalizations, such rhythms. The expressivity of the lived body implies a singular relation to others, and a different kind of intercorporeity than would be the case for two merely physical bodies. This intercorporeity should be understood as inter-temporality. Merleau-Ponty proposes this at the end of the chapter on perception in his Phenomenology of Perception, when he says,

But two temporalities are not mutually exclusive as are two consciousnesses, because each one knows itself only by projecting itself into the present where they can interweave.

Thus our bodies as different rhythms of duration can coexist and communicate, can synchronize to each other – in the same way that my body vibrated to the colors of the sensible world. But, in the case of two lived bodies, the synchronization occurs on both sides – with the result that I can experience an internal resonance with the other when the experiences harmonize, or the shattering disappointment of a  miscommunication when the attempt fails. The experience of coexistence is hence not a guarantee of communication or understanding, for this communication must ultimately be based on our differences as expressive bodies and singular durations. Our coexistence calls forth an attempt, which is the intuition.

Black Hole Complementarity: The Case of the Infalling Observer

The four postulates of black hole complementarity are:

Postulate 1: The process of formation and evaporation of a black hole, as viewed by a distant observer, can be described entirely within the context of standard quantum theory. In particular, there exists a unitary S-matrix which describes the evolution from infalling matter to outgoing Hawking-like radiation.

Postulate 2: Outside the stretched horizon of a massive black hole, physics can be described to good approximation by a set of semi-classical field equations.

Postulate 3: To a distant observer, a black hole appears to be a quantum system with discrete energy levels. The dimension of the subspace of states describing a black hole of mass M is the exponential of the Bekenstein entropy S(M).

We take as implicit in postulate 2 that the semi-classical field equations are those of a low energy effective field theory with local Lorentz invariance. These postulates do not refer to the experience of an infalling observer, but states a ‘certainty,’ which for uniformity we label as a further postulate:

Postulate 4: A freely falling observer experiences nothing out of the ordinary when crossing the horizon.

To be more specific, we will assume that postulate 4 means both that any low-energy dynamics this observer can probe near his worldline is well-described by familiar Lorentz-invariant effective field theory and also that the probability for an infalling observer to encounter a quantum with energy E ≫ 1/r_s (measured in the infalling frame) is suppressed by an exponentially decreasing adiabatic factor as predicted by quantum field theory in curved spacetime. We will argue that postulates 1, 2, and 4 are not consistent with one another for a sufficiently old black hole.

Consider a black hole that forms from collapse of some pure state and subsequently decays. Dividing the Hawking radiation into an early part and a late part, postulate 1 implies that the state of the Hawking radiation is pure,

|Ψ⟩= ∑_i |ψ_i⟩E ⊗|i⟩_L —– (1)

Here we have taken an arbitrary complete basis |i⟩_L for the late radiation. We use postulates 1, 2, and 3 to make the division after the Page time when the black hole has emitted half of its initial Bekenstein-Hawking entropy; we will refer to this as an ‘old’ black hole. The number of states in the early subspace will then be much larger than that in the late subspace and, as a result, for typical states |Ψ⟩ the reduced density matrix describing the late-time radiation is close to the identity. We can therefore construct operators acting on the early radiation, whose action on |Ψ⟩ is equal to that of a projection operator onto any given subspace of the late radiation.

To simplify the discussion, we treat gray-body factors by taking the transmission coefficients T to have unit magnitude for a few low partial waves and to vanish for higher partial waves. Since the total radiated energy is finite, this allows us to think of the Hawking radiation as defining a finite-dimensional Hilbert space.

Now, consider an outgoing Hawking mode in the later part of the radiation. We take this mode to be a localized packet with width of order r_s corresponding to a superposition of frequencies O(r⁻¹_s). Note that postulate 2 allows us to assign a unique observer-independent s lowering operator b to this mode. We can project onto eigenspaces of the number operator b^†b. In other words, an observer making measurements on the early radiation can know the number of photons that will be present in a given mode of the late radiation.

Following postulate 2, we can now relate this Hawking mode to one at earlier times, as long as we stay outside the stretched horizon. The earlier mode is blue-shifted, and so may have frequency ω_* much larger than O(r⁻¹_s) though still sub-Planckian.

Next consider an infalling observer and the associated set of infalling modes with lowering operators a. Hawking radiation arises precisely because

b = ∫₀^∞ dω B(ω)a_ω + C(ω)a^†_ω —– (2)

so that the full state cannot be both an a-vacuum (a|Ψ⟩ = 0) and a b^†b eigenstate. Here again we have used our simplified gray-body factors.

The application of postulates 1 and 2 has thus led to the conclusion that the infalling observer will encounter high-energy modes. Note that the infalling observer need not have actually made the measurement on the early radiation: to guarantee the presence of the high energy quanta it is enough that it is possible, just as shining light on a two-slit experiment destroys the fringes even if we do not observe the scattered light. Here we make the implicit assumption that the measurements of the infalling observer can be described in terms of an effective quantum field theory. Instead we could simply suppose that if he chooses to measure b^†b he finds the expected eigenvalue, while if he measures the noncommuting operator a^†a instead he finds the expected vanishing value. But this would be an extreme modification of the quantum mechanics of the observer, and does not seem plausible.

Figure below gives a pictorial summary of our argument, using ingoing Eddington-Finkelstein coordinates. The support of the mode b is shaded. At large distance it is a well-defined Hawking photon, in a predicted eigenstate of b^†b by postulate 1. The observer encounters it when its wavelength is much shorter: the field must be in the ground state a^†_ωa_ω = 0, by postulate 4, and so cannot be in an eigenstate of b^†b. But by postulate 2, the evolution of the mode outside the horizon is essentially free, so this is a contradiction.

Figure: Eddington-Finkelstein coordinates, showing the infalling observer encountering the outgoing Hawking mode (shaded) at a time when its size is ω⁻¹_* ≪ r_s. If the observer’s measurements are given by an eigenstate of a^†a, postulate 1 is violated; if they are given by an eigenstate of b^†b, postulate 4 is violated; if the result depends on when the observer falls in, postulate 2 is violated.

To restate our paradox in brief, the purity of the Hawking radiation implies that the late radiation is fully entangled with the early radiation, and the absence of drama for the infalling observer implies that it is fully entangled with the modes behind the horizon. This is tantamount to cloning. For example, it violates strong subadditivity of the entropy,

S_AB + S_BC ≥ S_B + S_ABC —– (3)

Let A be the early Hawking modes, B be outgoing Hawking mode, and C be its interior partner mode. For an old black hole, the entropy is decreasing and so S_AB < S_A. The absence of infalling drama means that S_BC = 0 and so S_ABC = S_A. Subadditivity then gives S_A ≥ S_B + S_A, which fails substantially since the density matrix for system B by itself is thermal.

Actually, assuming the Page argument, the inequality is violated even more strongly: for an old black hole the entropy decrease is maximal, S_AB = S_A − S_B, so that we get from subadditivity that S_A ≥ 2S_B + S_A.

Note that the measurement of N_b takes place entirely outside the horizon, while the measurement of N_a (real excitations above the infalling vacuum) must involve a region that extends over both sides of the horizon. These are noncommuting measurements, but by measuring N_b the observer can infer something about what would have happened if N_a had been measured instead. For an analogy, consider a set of identically prepared spins. If each is measured along the x-axis and found to be +1/2, we can infer that a measurement along the z-axis would have had equal probability to return +1/2 and −1/2. The multiple spins are needed to reduce statistical variance; similarly in our case the observer would need to measure several modes N_b to have confidence that he was actually entangled with the early radiation. One might ask if there could be a possible loophole in the argument: A physical observer will have a nonzero mass, and so the mass and entropy of the black hole will increase after he falls in. However, we may choose to consider a particular Hawking wavepacket which is already separated from the streched horizon by a finite amount when it is encountered by the infalling observer. Thus by postulate 2 the further evolution of this mode is semiclassical and not affected by the subsequent merging of the observer with the black hole. In making this argument we are also assuming that the dynamics of the stretched horizon is causal.

Thus far the asymptotically flat discussion applies to a black hole that is older than the Page time; we needed this in order to frame a sharp paradox using the entanglement with the Hawking radiation. However, we are discussing what should be intrinsic properties of the black hole, not dependent on its entanglement with some external system. After the black hole scrambling time, almost every small subsystem of the black hole is in an almost maximally mixed state. So if the degrees of freedom sampled by the infalling observer can be considered typical, then they are ‘old’ in an intrinsic sense. Our conclusions should then hold. If the black hole is a fast scrambler the scrambling time is r_s ln(r_s/l_P), after which we have to expect either drama for the infalling observer or novel physics outside the black hole.

We note that the three postulates that are in conflict – purity of the Hawking radiation, absence of infalling drama, and semiclassical behavior outside the horizon — are widely held even by those who do not explicitly label them as ‘black hole complementarity.’ For example, one might imagine that if some tunneling process were to cause a shell of branes to appear at the horizon, an infalling observer would just go ‘splat,’ and of course Postulate 4 would not hold.