## Quantum fluctuations in geometry as a new kind of noise?
The news of yesterday morning came in email from Jack Sarfatti. The news was that gravitational detectors in GEO600 experiment have been plagued by unidentified noise in the frequency range 300-1500 Hz. Craig J. Hogan has proposed an explanation in terms of holographic Universe. By reading the paper I learned that assumptions needed are essentially those of quantum TGD. Light-like 3-surfaces as basic objects, holography, effective 2-dimensionality, are some of the terms appearing repeatedly in the article. Maybe this means a new discovery giving support for TGD. I hope that it does not make my life even more difficult in Finland. Readers have perhaps noticed that the discovery of new longlived particle in CDF predicted by TGD already around 1990 turned out to be one of most fantastic breakthroughs of TGD since the reported findings could be explained at quantitative level. The side effect was that Helsinki University did not allow me to use the computer for homepage anymore and they also refused to redirect visitors to my new homepage. The goal was achieved: I have more or less disappeared from the web. It seems that TGD is becoming really dangerous and power holds of science are getting nervous. In any case, I could not resist the temptation to spend the day with this problem although I had firmly decided to use all my available time to the updating of basic chapters of quantum TGD.
Consider first the graviton detector used in GEO600 experiment. The detector consists of two long arms (the length is 600 meters)- essentially rulers of equal length. The incoming gravitational wave causes a periodic stretch of the arms: the lengths of the rulers vary. The detection of gravitons means that laser beam is used to keep record about the varying length difference. This is achieved by splitting the laser beam into two pieces using a beam splitter. After this the beams travel through the arms and bounce back to interfere in the detector. Interference pattern tells whether the beam spent slightly different times in the arms due to the stretching of arm caused by the incoming gravitational radiation. The problem of experimenters has been the presence of an unidentified noise in the range 100-1500 Hz. The prediction of Measurement of quantum fluctuations in geometry by Craig Hogan published in Phys. Rev. D 77, 104031 (2008) is that holographic geometry of space-time should induce fluctuations of classical geometry with a spectrum which is completely fixed . Hogan's prediction is very general and - if I have understood correctly - the fluctuations depend only on the duration (or length) of the laser beam using Planck length as a unit. Note that there is no dependence on the length of the arms and the fluctuations characterize only the laser beam. Although Planck length appears in the formula, the fluctuations need not have anything to with gravitons but could be due to the failure of the classical description of laser beams. The great surprise was that the prediction of Hogan for the noise is of same order of magnitude as the unidentified noise bothering experiments in the range 100-700 Hz.
Let us try to understand Hogan's theory in more detail.
- The basic quantitative prediction of the theory is very simple. The spectral density of the noise for high frequencies is given by h
_{H}= t_{P}^{1/2}, where t_{P}=(hbar G)^{1/2}is Planck time. For low frequencies h_{H}is proportional to 1/f just like 1/f noise. The power density of the noise is given by t_{P}and a connection with poorly understood 1/f noise appearing in electronic and other systems is suggestive. The prediction depends only Planck scale so that it should very easy to kill the model if one is able to reduce the noise from other sources below the critical level t_{P}^{1/2}. The model predicts also the the distribution characterizing the uncertainty in the direction of arrival for photon in terms of the ratio l_{P}/L. Here L is the length or beam of equivalently its duration. A further prediction is that the minimal uncertainty in the arrival time of photons is given by Δ t= (t_{P}t)^{1/2}and increases with the duration of the beam. - Both quantum and classical mechanisms are discussed as an explanation of the noise. Gravitational holography is the key assumption behind both models. Gravitational holography states that space-time geometry has two space-time dimensions instead of three at the fundamental level and that third dimension emerges via holography. A further assumption is that light-like (null) 3-surfaces are the fundamental objects. Sounds familiar!
2.1 Heuristic argument
The model starts from an optics inspired heuristic argument.
- Consider a light ray with length L, which ends to aperture of size D. This gives rise to a diffraction spot of size λL/D. The resulting uncertainty of the transverse position of source is minimized when the size of diffraction spot is same as aperture size. This gives for the transverse uncertainty of the position of source Δ x= (λ L)
^{1/2}. The orientation of the ray can be determined with a precision Δ θ= (λ/L)^{1/2}. The shorter the wavelength the better the precision. Planck length is believed to pose a fundamental limit to the precision. The conjecture is that the transverse indeterminacy of Planck wave length quantum paths corresponds to the quantum indeterminacy of the metric itself. What this means is not quite clear to me. - The basic outcome of the model is that the uncertainty for the arrival times of the photons after reflection is proportional to
Δ t =t _{P}^{1/2}× (sin(θ))^{1/2}×sin(2θ),where θ denotes the angle of incidence on beam splitter. In normal direction Δ t vanishes. The proposed interpretation is in terms of Brownian motion for the distance between beam splitter and detector the interpretation being that each reflection from beam splitter adds uncertainty. This is essentially due to the replacement of light-like surface with a new one orthogonal to it inducing a measurement of distance between detector and bean splitter.
This argument has some aspects which I find questionable.
- The assumption of Planck wave length waves is certainly questionable. The underlying is that it lead to the classical formula involving the aperture size which is eliminated from the basic formula by requiring optimal angular resolution. One might argue that a special status of waves with Planck wave length breaks Lorentz invariance but since the experimental apparatus defines a preferred coordinate system this ned not be a problem.
- Unless one is ready to forget the argument leading to the formula for Δ θ, one can argue that the description of the holographic interaction between distant points induced by these Planck wave length waves in terms of aperture with size D= (l
_{P}L)^{1/2}should have some more abstract physical counterpart. Could elementary p"/public_html/articles/ as extended 2-D objects (as in TGD) play the role of ideal apertures to which a radiation with Planck wave length arrives? If one gives up the assumption about Planck wave radiation the uncertainty increases as λ. To my opinion one should be able to deduced the basic formula without this kind of argument.
Second argument can do without diffraction but still uses Planck wave length waves.
- The interactions of Planck wave length radiation at null surface at two different times corresponding to normal coordinates z
_{1}and z_{2}at these times are considered. From the standard uncertainty relation between momentum and position of the incoming particle one deduces uncertainty relation for transverse position operators x(z_{i}), i=1,2. The uncertainty comes from uncertainty of x(z_{2}) induced by uncertainty of the transverse momentum p_{x}(z_{i}). The uncertainty relation is deduced by assuming that (x(z_{2})-x(z_{1})/(z_{2}-z_{1}) is the ratio of transversal and longitudinal wave vectors. This relates x(z_{2}) to p_{x}(z_{i}) and the uncertainty relation can be deduced. The uncertainty increases linearly with z_{2}-z_{1}. Geometric optics is used to describing the propagating between the two points and this should certainly work for a situation in which wavelength is Planck wavelength if the notion of Planck wave length wave makes sense. From this formula the basic predictions follow. - Hogan emphasizes that the basic result is obtained also classically by assuming that light-like surfaces describing the propagation of light between ends points of arm describe Brownian like random walk in directions transverse to the direction of propagation. I understand that this means that Planck wave length wave is not absolutely necessary for this approach.
Hogan discusses also an effective description of holographic noise in terms of gravitational wave packet passing through the system.
- The holographic noise at frequency f has equivalent description in terms of a gravitational wave packet of frequency f and duration T=1/f passing through the system. In this description the variance for the length difference of arms using standard formula for gravitational wave packet
Δl ^{2}/l^{2}= h^{2}f ,where h characterizes the spectral density of gravitational wave. - For high frequencies one obtains
h= h _{P}= (t_{P})^{1/2}. - For low frequencies the model predicts
h= (f _{res}/f)(t_{P})^{1/2}.Here f _{res}characterized the inverse residence time in detector and is estimated to be about 700 Hz in GEO600 experiment. - The predictions of the theory are compared to the unidentified noise in the frequency range 100-600 Hz which introduces amplifying factor varying from 7 to 1. The orders of magnitude are same.
In TGD based model for the claimed noise on can avoid the assumption about waves with Planck wave length. Rather Planck length corresponds to the transversal cross section of so called massless extremals (MEs) assignable to MEs and orthogonal to the direction of propagation. Further elements are so called number theoretic braids leading to the discretization of quantum TGD at fundamental level. The mechanism inducing the distribution for the travel times of reflected photon is due to the transverse extension of MEs, discretization in terms of number theoretic braids. Note that also in Hogan's model it is essential that one can speak about position of particle in the beam.
Consider first the general picture behind the TGD inspired model.
- What authors emphasize can be condensed to the following statement:
*The transverse indeterminacy of Planck wave length seems likely to be a feature of 3+1 D space-time emerge is as a dual of quantum theory on a 2+1-D null surface*. In TGD light-like 3-surfaces indeed are the fundamental objects and 4-D space-time surface is in a holographic relation to these light-like 3-surfaces. The analog of conformal invariance in light-like radial direction implies that partonic 2-surfaces are actually basic objects in short scales in the sense that one 3-dimensionality only in discretized sense. - Both the interpretation as almost topological quantum field theory, the notion of finite measurement resolution, number theoretical universality making possible p-adicization of quantum TGD, and the notion of quantum criticality lead to a fundamental description in terms of discrete points sets. These are defined as intersections of what I call number theoretic braids with partonic 2-surfaces X
^{2}at the boundaries of causal diamonds identified as intersections of future and paste directed light-cones forming a fractal hierarchy. These 2-surfaces X^{2}correspond to the ends of light-like three surfaces. Only the data from this discrete point set is used in the definition of M-matrix: there is however continuum of selections of this data set corresponding to different directions of light-like ray at the boundary of light-cone, and in detection one of these direction is selected and corresponds to the direction of beam in the recent case. - Fermions correspond to CP
_{2}vacuum extremal with Euclidian signature of induced metric condensed to space-time sheet with Minkowskian signature and light-like wormhole throat for which 4-metric is degenerate carries the quantum numbers. Bosons correspond to wormhole contacts consisting of a piece of CP_{2}vacuum extremal connecting two two space-time sheets with Minkowskian signature of induced metric. The strands of number theoretic braids carry fermionic quantum numbers and discretization is interpreted as a space-time correlate for the finite measurement resolution implying the effective grainy nature of 2-surfaces.
Consider now the TGD inspired model for a laser beam of fixed duration T.
- In TGD framework the beams of photons and perhaps also photons themselves would have so called massless extremals as space-time correlates. The identification of gauge bosons as wormhole contacts means that there is a pair of MEs connected by a pieces of CP
_{2}type vacuum extremal and carrying fermion and antifermion at the wormhole throats defining light-like 3-surfaces. The intersection of ME with light-cone boundary would represent partonic 2-surface and any transverse cross section of the M^{4}projection of ME is possible. - The reflection of ME has description in terms of generalized Feynman diagrams for which the incoming lines correspond to the light-like three surfaces and vertices to partonic 2-surfaces at which the MEs are glued together. In this simplest model this surface defines transverse cross section of both incoming and outgoing ME. The incoming and outgoing braid strands end to different points of the cross section because if two points coincide the N-point correlation function vanishes. This means that in the reflection the distribution for the positions of braid points representing exact positions of photon change in non-deterministic manner. This induces a quantum distribution of transverse coordinates associated with braid strands and in the detection state function reduction occurs fixing the position of braid strands.
- The transversal cross section has maximum area when it is parallel to ME. In this case the area is apart from a numerical constant equal to d×L, L the length defined by the duration of laser beam defining the length of ME and d the diameter of orthogonal cross section of ME. This makes natural the assumption about Gaussian distribution for the positions of points in the cross section as Gaussian with variance equal to d×L. The distribution proposed by Hogan is obtained if d is given by Planck length. This would mean that the minimum area for a cross section of ME is very small, about S=hbar×G. This might make sense if the ME represents laser beam.
- The assumption susceptible to criticism is that for the primordial ME representing photon the area of cross section orthogonal to the direction of propagation is assumed to be always given by Planck length. This assumption of course replaces Hogan's Planck wave. Note that the classical four-momentum of ME is massless. One could however argue that in quantum situation transverse momentum square is well defined quantum number and of order Planck mass mass squared.
- In TGD Universe single photon would differ from infinitely narrow ray by having thickness defined by Planck length. There would be just single braid strand and its position would change in the reflection. The most natural interpretation indeed is that the pair of space-time sheets associated with photon consists of MEs with different transversal size scales: larger ME could represent laser beam. The noise would come from the lowest level in the hierarchy. One could argue that the natural size for M
^{4}projection of wormhole throat is of order CP_{2}size R and therefore roughly 10^{4}Planck lengths. If the cross section has area of order R^{2}, where R is CP_{2}size, the spectral density would be roughly by a factor 100 larger than for Planck length and this might predict too large holographic noise in GEO600 experiment if the value of f_{res}is correct. The assumption that the Gaussian characterizing the position distribution of the wormhole throat is very strongly concentrated near the center of ME with transverse size given by R looks un-natural. - It is important to notice that single reflection of primordial ME corresponds to a minimum spectral noise. Repeated reflections of ME in different directions gradually increase the transversal size of ME so that the outcome is cylindrical ME with radius of order L =cT, where T is the duration of ME. At this limit the spectral density of noise would be T
^{1/2}meaning that the uncertainty in the frequency assignable to the arrival time of photons would of same order as the oscillation period f=1/T assignable to the original ME. The interpretation is that the repeated reflections gradually generate noise and destroy the coherence of the laser beam. This would however happen at single particle level rather than for a member of fictive ensemble. Quite literally, photon would get old! This interpretation conforms with the fact that in TGD framework thermodynamics becomes part of quantum theory and thermodynamical ensemble is represented at single particle level in the sense and time like entanglement coefficients between positive and negative energy parts of zero energy state define M-matrix as a product of square root of diagonal density matrix and of S-matrix. - The notion of number theoretic braid is essential for the interpretation for what happens in detection. In detection the positions of ends of number theoretic braid are measured and this measurement fixes the exact time spent by photons during the travel. Similar position measurement appears also in Hogan's argument. Thus the overall picture is more or less same as in the popular representation where also the grainy nature of space-time is emphasized.
- I already mentioned the possible connection with poorly understood 1/f noise appearing in very many systems. The natural interpretation would be in terms of MEs.
It is interesting to combine this picture with the vision about the hierarchy of Planck constants (I am just now developing in detail the representation of the ideas involved from a perspective given by the intense work during last five years).
- If one accepts that dark matter corresponds to a hierarchy of phases of matter labeled by a hierarchy of Planck constants with arbitrarily large values, one must conclude that Planck length l
_{P}proportional to hbar^{1/2}, has also spectrum. Primordial photons would have transversal size scalings as hbar^{1/2}. One can consider the possibility that for large values of hbar the transversal size saturates to CP_{2}length R ≈10^{4}× l_{P}. The spectral density of the noise would scale as hbar^{1/4}at least up to the critical value hbar_{cr}=R^{2}/G, which is in the range [2.3683, 2.5262]× 10^{7}. The preferred values of hbar number theoretically simple integers expressible as product of distinct Fermat primes and power of 2. hbar_{crit}/hbar_{0}=3× 2^{23}is integer of this kind and belongs to the allowed range of critical values. - The order of magnitude for gravitational Planck constant assignable to the space-time sheets mediating gravitational interaction is gigantic - of order hbar
_{gr}≈ GM^{2}- so that the noise assignable to gravitons would be gigantic in astrophysical scales unless R serves as the upper bound for the transverse size of both primordial gauge bosons and gravitons. - If ordinary photonic space-time sheets are in question hbar has its standard value. For dark photons which I have proposed to play a key role in living matter, the situation changes and Δl
^{2}/l^{2}would scale like hbar^{1/2}at least up to critical value of Planck constant. Above this value of Planck constant spectral density would be given by R and Δl^{2}/l^{2}would scale like R/l and Δ θ like (R/l)^{1/2}.
For details and background see the updated chapter Quantum Astrophysics of "Physics in Many-Sheeted Space-time". |