Tuesday, October 22, 2013

Is space and time emergent? ER-EPR correspondence adds a voice

After a cold September and early October, an Indian summer finally arrived to Central Europe. Well, it's only called "Indian summer" in Northern America. In Europe, we have a different meteorological phenomenon, the grandma summer (at least in Czech we call it this way – because the floating spider webs resemble grandmas' grey hair) ;-), and it's disputable whether the grandma summer and the Indian summer may be viewed as equivalents. Europe and Northern America are allowed to differ in other respects than just the human history and politics.

Such a nice weather keeps one offline. But another reason was a repulsion from an overwhelming avalanche of anti-quantum zealots who filled my very own blog. Everyone who knows me can confirm that I am holier than Jesus Christ, trying to self-sacrifice as much as I can, hurt no one, and so on. But I returned back to my senses an hour ago. This is my blog, in fact, my parttime job of a sort, so I am responsible for the management and maintenance here. A comment thread with 180+ comments from which those who should learn something don't learn anything at all is just too huge a waste of time (and space).
Related and hopefully free of misinteractions: Andreas Karch (a TRF guest blogger, among more important things) just released a nice text about the black hole interior for the APS. I endorse the content and it's similar e.g. to this text of mine.
James Gallagher was placed on the blacklist because he's been such a pain in the neck (we have also heard that John Bell was a greater mind and made greater contributions than John von Neumann – holy cow, how insane someone has to be to believe a similar thing) and I will do the same thing with analogous lunatics much more quickly than I was doing it before because being repelled from one's own blog isn't how things should work. The ban(s) may be temporary but this feeling of being overwhelmed with the repetitive nonsense is something I am likely to remember at least for a week.

A realist like me finds it deeply discouraging to write about cutting-edge research and the most refined ideas that could have been crisply formulated only at the beginning of the 21st century (or the end of the 20th century) if I am being constantly assured that 95+ percent of the readers still fail to understand quantum mechanics that's been settled for more than 85 years. Among the remaining 5- percent of the readers, 95+ percent of them probably still fail to understand the insight that string theory is the only known consistent theory of quantum gravity. Of course, to understand cutting-edge correct ideas about quantum gravity, one has to follow many more things than just two "Yes" answers to quantum mechanics and basic string theory.

Are there any readers left after those filters are imposed? Does it make any sense to write about advanced scientific topics? Isn't the feeling of disappointment and disgust inevitable if one is throwing these ideas to the broad public? I am afraid that the bad outcomes are inevitable. Cutting-edge theoretical physics doesn't mix well with the public because a vast majority of the public is composed of morons, at least according to the standards of the cutting-edge theoretical physics.

Apparently independently of each other, Sean Carroll and a German blogger ("What Is Viscosity...") revisit the question whether the time is fundamental. None of them says anything interesting or novel or modern (or correct, for that matter) and a German blogger only discusses a standard type of a wrong paper (the sort of paper that claims that quantum gravity requires Lorentz violation) but I decided to write something hopefully more meaningful about these matters, too.

The newest part of this blog post will be my interpretation of what the ER-EPR correspondence by Maldacena and Susskind seems to imply about the question whether the space and time are fundamental.

The first TRF text about the emergent space and emergent time was posted in October 2004, when this blog was less than one week old.

Quantum foam

A confused German physicist and Carroll

First, let me mention things I am not going to talk about because they're self-evidently wrong.

I won't be talking about cosmological mechanisms behind the arrow of time because there aren't any. The thermodynamic arrow of time (and other equivalent arrows of time) may be shown to arise from the logical arrow of time. The laws of physics allow us to determine "predictions" out of "assumptions". By definition, the laws of physics dictate how the future evolves from the past. So all the unambiguously calculable (= by the laws of physics unambiguously determined) probabilities assume that the past is known and the future is not. It follows that the probabilities in the opposite situation (future/present is known but the present/past is not) are not unambiguously calculable because they depend on priors etc.

If the laws of physics involve the time coordinate, it must agree with the existence of a chronology, an ordering of events that influence each other. The laws of special relativity guarantee that the ordering between events A,B is the same regardless of the reference frame whenever A,B may influence each other. To make this happen, relativity prohibits the superluminal communication.

However, it may be easily calculated – and it's been calculated 17 times on this blog – that the probability of a decreasing-entropy evolution \(B^*\to A^*\) isn't "equally high" as the probability of the entropy-increasing evolution \(A\to B\), as brutally confused people like Carroll like to say. (The asterisk stands for the CPT conjugation or just the simple T reversal when the latter is a symmetry of the theory.) Instead, the decreasing-entropy process may be seen to be less likely by an exponentially huge factor \(\exp[(S_B-S_A)/k]\). The asymmetry is due to the "summing over the final microstates but averaging over initial microstates" which is inseparable from the logical arrow of time and boils down to nothing else than the elementary probability calculus (rules that were known long before quantum mechanics).

I won't talk about the German physicist's misconceptions, either. Among other things, he or she believes that quantum gravity should produce some Lorentz-symmetry-violating terms and follows wrong papers trying to investigate the same claim. But relativity prohibits such things in all phenomena in the world, even in quantum gravity – in fact, especially in quantum gravity. The Lorentz symmetry is a principle of physics that Einstein was able to "see" through the basic empirical data available to all of us (you really can't tell whether a train is moving if it is moving smoothly). The Lorentz symmetry constrains the "final product", the outcome of good theories' calculations including all corrections. It doesn't apply "just" to some zeroth-order approximations in theories; in fact, zeroth-order approximations may break the Lorentz symmetry. But when the relevant corrections are added etc., the symmetry has to be restored in the final product.

The idea of a Lorentz violation in quantum gravity is particularly silly because the word "gravity" in "quantum gravity" actually means general relativity and general relativity is obliged to preserve the Lorentz symmetry (which may be spontaneously broken if the spacetime isn't empty, at various scales etc.). If the symmetry is broken at the Planck scale, it will be broken at any scale and one will find herself in the conflict not only with the principles that Einstein was able to guess by remembering his experiences in the train but also with much more accurate experimental tests of the symmetry: if the maximum speed that different particles may achieve differ at the Planck distance scale, they will differ at any scale!

Quantum foam: well-known geometry doesn't have to be a part of a theory of quantum gravity

The character of the modifications of geometry that quantum gravity requires is often being misinterpreted and distorted and lots of clarifications have to be stressed to listeners who have been brainwashed by loads of crap by writers like Smolin or a former German co-author of his.

First, those people want to believe you that the new quantum gravity or Planckian phenomena start to operate when the coordinate distances (or time coordinate differences) become comparable to the Planck length (or Planck time). But that definitely doesn't happen. Relativity implies that coordinates don't have objective physical consequences; their shortness (and value) is relative – they depend on the observer.

New quantum gravity phenomena only occur when invariant quantities with the units of length or time, e.g. the proper lengths or the radii of objects measured in their rest frame (or the impact parameter measured in the center-of-mass frame), drop to values comparable to the Planck length (or Planck time). For example, a black hole whose Schwarzschild radius approaches the Planck length marginally ceases to obey the usual laws of black holes in classical general relativity; that's where the black holes start to intensely feel quantum mechanics and their microstates don't differ "qualitatively" from generic unstable particles such as Higgs bosons. Black holes smaller than the Planck length don't really exist.

On the other hand, when coordinate differences are sub-Planckian, it doesn't mean anything new. For example, a photon's wavelength may be 100 times shorter than the Planck length and this photon obeys all the usual properties you expect from relativity. After all, in a different inertial frame, the photon's wavelength is much longer than the Planck scale (due to the Doppler shift).

We expect all the subleading corrections to dynamics – but only those that don't violate the universally valid symmetries such as the Lorentz symmetry – to be comparably large to the usual "leading" terms when the typical distance scale of the physical phenomena (such as the curvature radius or the wavelength of colliding gravitational waves: it's necessary for them to be colliding for the wavelength's shortness to have an invariant character) approaches the Planck scale. For example, the Einstein-Hilbert action that generates Einstein's equations of general relativity is \[

S_{EH} =\frac{1}{16\pi G} \int \dd^d x\, R\sqrt{-|g|}

\] where \(R\) is the Ricci scalar and \(|g|\), often written simply as \(g\), is the determinant of the metric tensor (which is negative). The action above is an approximation and a more accurate formula includes corrections of higher order in the Riemann curvature tensor. For example, if we add the curvature-squared terms, we get something like\[

S_{EH} =\frac{1}{16\pi G} \int \dd^d x\, (R+ c R^2+\dots)\sqrt{-|g|}

\] where \(R^2\) may be the squared Ricci scalar but also other contractions of two copies of the Riemann tensor. What is the magnitude of \(c\)? It's the point of the Planck units that in the pure gravity case, all constants of physics are expected to be of order one in the Planck units. Because \(R^2\) has units of \({\rm length}^{-2}\) times the units of \(R\) (the curvature contains the second spacetime derivatives of the metric, and otherwise just the dimensionless metric), it's clear that \(c\) has units of \({\rm length}^{+2}\) to compensate the extra second negative power from the extra \(R\) in \(R^2\). By dimensional analysis, \(c\) must be of order \[

c\sim \text{Planck length}^{+2}.

\] When \(R\) is of order \(\text{Planck length}^{-2}\) as well, we see that the terms \(R\) and \(cR^2\) are of the same order. The Einstein-Hilbert action (and the usual term, the Einstein tensor, that it generates in the equations of motion) is no longer a good approximation for the dynamics. The subleading terms are of the same order, \(R\sim cR^2\), and the same is true for arbitrarily higher-order terms (and even the nonperturbative corrections).

So once the distances in the physical problem (e.g. the curvature radius) become comparable to the Planck length, the usual approximations we're used to from the long-distance experience (e.g. Einstein's equations) cease to be good and everything compatible with the universal principles (and symmetries) of physics that may happen generally will happen. But that doesn't mean that any random science-fiction crazy idea will happen.

Off-topic: Does this really happen to a neodymium magnet in FAT copper pipe?

For example, it doesn't mean that there is anything discrete about the geometry at the Planck scale. The word "quantum" in "quantum mechanics" was historically chosen because certain quantities (like the energy carried by photons in the electromagnetic field or the energy of the electron in the hydrogen atom) are discrete. But quantum mechanics is more general than that and many quantities (especially the position but also the momentum in a noncompact space or the energy of a free particle or an electron in/outside an ionized atom) are allowed to have and love to have a continuous spectrum. The spacetime can't be discrete in the most naive sense because such a situation would massively contradict the Lorentz invariance – and its very accurate empirical tests.

So you should imagine that all expectations at the Planck scale are corrected by corrections of order 100 percent (if the curvature radius etc. of the problem is comparable to the Planck scale) but certain things may stay [thanks, Peter] "qualitatively the same" while other things may change qualitatively. Quantum gravity doesn't imply that everything must change "qualitatively"; it doesn't imply that everything stays qualitatively the same, either.

John Wheeler promoted the concept of the "quantum foam" that is pictured on the image above (I don't mean the image of the Indian summer but the next one). If we measure \(x\) in the ground state of a quantum harmonic oscillator, we don't get zero. Instead, we get a number that is normally distributed around zero where the width of the normal distribution is given by the width of the Gaussian (squared ground state wave function).

Similarly, if we measure the shape of the geometry with the Planckian accuracy, we won't get a flat space. Instead, we get the wildly fluctuating quantum foam that may even contain the wormholes, as the pictures often indicate. Before string theory, there hasn't been any formalism that would allow us to "derive" whether or not the wormholes and other topologically nontrivial modifications are really there. String theory is powerful enough to answer these questions. However, we will mostly talk about some considerations that hold in string theory but they don't depend on its "stringiness" much. I primarily mean the ER-EPR correspondence.

We want to know: Are those bridges there in the Planckian landscape? Can we talk about the shape of the geometry at all when the resolution is this fine? Is the geometry and topology of the space a well-defined property of it? People have been asking similar questions – usually in a less careful form than my questions – for decades. Just two decades or so ago, physicists began to answer these questions. The recent findings by Susskind and Maldacena brought a new twist to all these results.

T-duality and other geometry-changing operations in string theory

The correct answer to one of the questions above is that
[A] consistent quantum theory of gravity isn't obliged to treat the metric tensor (and the exact shape of the geometry) as a good degree of freedom at the Planckian resolution.
Only when the distances are much longer than the Planck scale, it becomes possible to isolate the "usual classical geometry" and the leading terms associated with it, such as the Einstein-Hilbert action, are much larger than all the corrections. So the physical phenomena "mostly" follow the usual rules of phenomena in a spacetime.

At the Planck scale, any of these expectations may break down. But I think it's important to stress that they don't have to break down. The theory – e.g. string theory (which is really the only consistent example but let me not erase the letters "e.g.", anyway) – doesn't have to agree with the expectation that the metric tensor is a good observable at the Planck scale (and doesn't have to answer questions about "its" values because "it" doesn't necessarily exist). On the other hand, string theory is not obliged to agree with the proposition that the metric tensor doesn't exist, either!

Similar comments hold about locality. Locality may be broken when the proper distances become Planckian but nothing forbids the possibility that with an appropriate field redefinition, the locality will actually hold exactly. So the breaking of the "existence of the metric tensor" and the "violations of the locality" are allowed but when they occur, they may conceivably be mere artifacts of the choice of field redefinitions, too.

The AdS/CFT correspondence describes all of type IIB string theory (supergravity, strings, branes, black holes etc.) in the Anti de Sitter (AdS) bulk in terms of a conformal field theory (CFT) on the boundary of the AdS space at infinity. The CFT is a theory analogous QCD, at least in the most famous cases in \(d=4\), and doesn't "obviously" contain anything that would look like \(g_{\mu\nu}(\rho,x,y,z,t)\), the metric tensor in \(d=5\). However, there's no immediate proof that such observables (the dynamical metric field in the bulk) can't be extracted from the CFT. We usually assume that it can't be done but there's really no known principle that would imply that it can't be done.

Does quantum gravity and/or string/M-theory (which is really the same thing, at the end) allow us to interpret any physical phenomena as phenomena that take place in a well-defined geometric background? What's the shape and topology of this background?

That's a good question, not only because it is mine and I carefully eliminated all known hidden invalid assumptions from similar questions that others could ask.

If you think about the answer, you should realize that even though a theory of quantum gravity isn't obliged to include the metric tensor at the Planck scale among the "good quantum numbers" (well-defined degrees of freedom) at the Planck scale, the answer actually tends to be "Yes" in string theory.

In fact, string/M-theory has the tendency to allow us to interpret the same physical phenomena as physical phenomena that take place on many conceivable spacetime backgrounds, often very different spacetime backgrounds. What do I mean?

First, a gravitational wave is a particular curved configuration of the metric tensor (field) in classical general relativity. In a quantum theory, it is identified with a coherent state of many gravitons in the same one-particle state. One may build the gravitational wave – a curved spacetime metric – by adding particles (gravitons) into a flat spacetime. String theory makes this construction explicit and fully consistent (even when you ask any question about the gravitons' interactions).

Even though string theory wasn't obliged to provide you with a well-defined spacetime background for a given physical process, it did provide you with a well-defined spacetime background. In fact, infinitely many of them. Because a deformed spacetime may be constructed by adding graviton excitations into an undeformed (or any other deformed) spacetime, all physical phenomena in a topologically trivial spacetime may be interpreted as phenomena in any particular spacetime with the same topology (but different curvatures) equipped with different ensembles of particle-like excitations on top of the background.

Of course that we prefer the description that makes the spacetime look more empty (it is often more practical to talk about an empty curved classical spacetime than a flat spacetime with lots of gravitons in it) so one of the infinitely many descriptions of the spacetime may be more useful in practice (or more "natural" if the word "natural" is defined pragmatically). But at the fundamental level, all the spacetime geometries are equally good.

Dualities provide us with another example of the assertion that string/M-theory actually allows you to associate many background spacetimes with a given underlying process. The T-duality completely changes our ideas about the identity of the compactified circular dimensions. In one of the T-dual (and therefore mutually equivalent) descriptions, the compact dimension \(X^{10}\) is complementary to the momentum \(P^{10}\) of the strings or other objects. But there is another description in which the compact dimension is \(\tilde X^{10}\) which is complementary to \(\tilde P^{10}=W^{10}\), the winding number of strings around the original dimension \(X^{10}\).

These two directions, \(X^{10}\) and \(\tilde X^{10}\), aren't fully compatible with one another. We can't imagine that the strings are moving in a spacetime that has both of these dimensions, \(X^{10}\) and \(\tilde X^{10}\), at least not strings that are moving and winding in the usual largely unconstrained way. However, both descriptions, one with \(X^{10}\) and one with \(\tilde X^{10}\), are equally valid, equally legitimate, and offer us equally accurate frameworks to calculate all the physical predictions.

A special kind of T-duality is the mirror symmetry – which may be interpreted as a T-duality "reverting" the three coordinates of the \(T^3\) fibers if the Calabi-Yau manifold is written as a fibrations – and it implies that two topologically different Calabi-Yau manifolds produce the same physics. For example, some physical phenomena in type IIA string theory may be thought of as taking place in a spacetime with one Calabi-Yau manifold whose Hodge numbers are \((h^{1,1},h^{1,2})\). There exists an exactly equivalent description of these phenomena as type IIB string theory phenomena on a topologically different Calabi-Yau manifold with the reverted Hodge numbers \((h^{1,2},h^{1,1})\). None of them is better than the other. None of the geometries is "more real" than the other (at least when all the distance scales are comparable to the string length – which is the self-dual radius for a T-duality; it's not the Planck length). Both of them work but the dimensions (at least 3 of the 6 dimensions) in the two descriptions are incompatible with each other. They're different spaces. They're not projections of the same 9-dimensional or 12-dimensional space, for example.

The S-duality only acts on "internal" properties of the particles (light excitations and monopoles are interchanged, for example), but it preserves the spacetime in which they are embedded. However, U-dualities – more general duality transformations that act on internal as well as geometric quantities, i.e. they combine the actions known from S-dualities and T-dualities – do switch the background to a different one. That's also the case of the \(E_{k(k)}(\ZZ)\) discrete exceptional non-compact U-duality symmetries in M-theory on tori (with maximally supersymmetric supergravity at low energies). The compactified dimensions in all the infinitely many spacetimes related by these U-duality transformations are different. The momenta in the compact directions are equal to the M2-brane and M5-brane winding numbers (and more complicated charges) in the U-dual, equivalent descriptions (or their linear combinations).

The string-string duality (the heterotic-K3 duality) is yet another example of the fact that string theory often provides us with numerous exactly equivalent descriptions of the same physical phenomena that may disagree – even completely disagree – about the identity of the spacetime background, even about its topology. The AdS/CFT correspondence in particular or holography in general is yet another example in which the two equivalent descriptions may disagree about the number of spacetime dimensions, too. To ask whether the radial holographic dimension "exists" means to ask the question whether you prefer the AdS or the CFT description. If you're impartial enough, you know that you shouldn't take sides. They're equivalent.

In most of the interesting stringy cases, we're not dealing with two descriptions among which one of them is much more accurate (or much more symmetric) than the other and the other is just an approximation (or a contrived broken phase). Instead, both of the dual (equivalent) descriptions are equally accurate, they often have equal numbers of dimensions, they often have comparable symmetries (my favorite example is the \(SO(32)\) and \(E_8\times E_8\) heterotic string theory that are T-dual to each other if one correctly adjusts the Wilson lines and other parameters; one can't say that one of the gauge groups is a broken version of the other one – in fact, both groups have rank \(16\) and dimension \(496\)); there's no way to present one of them as "superior" in any way. In some sense, they are connected with different but equally good classical limits of the full quantum theory.

So I have mentioned many examples of insights, dualities, and constructions in string/M-theory that seem to support the same general story: Even though string/M-theory isn't obliged to provide us with a spacetime interpretation of physical phenomena, it always tends to give us a spacetime background for any phenomena that may occur. In fact, it's not just one spacetime but many (and because of the coherent graviton states we started with, and also because of the infinite order of the non-compact U-duality groups in M-theory, infinitely many) spacetime frameworks for any events.

At the beginning, we were warned that quantum gravity had the right to make all of our concepts and ideas about the spacetime background meaningless once we demand the Planckian resolution. For given physical processes in the quantum gravity regime, it could happen that no spacetime background was OK. But instead, the final result seems to paint a different story. It is possible to use a spacetime background – in fact, one may do so in many ways. In some sense, any event may be described as a more or less convoluted rearrangement of particles on any spacetime background.

Possibility to fix the gauge (light-cone gauge) in string theory & allowed topology change & ER-EPR

When you looked at the picture of the quantum foam at the top, you could have been afraid that these wormholes that may appear at the Planck scale will prevent us from describing a generic accurate nonperturbative process in quantum gravity as a process that occurs on a topologically trivial background. After all, doesn't the picture make it obvious that the right spacetime's topology is not trivial which should mean that every description in terms of a trivial topology must be wrong – or at least inaccurate?

As the previous long section suggested, the answer may be surprising but it shows the tremendously strong muscles of string/M-theory. The answer is that string theory allows you to use the background you choose, even the flat one, and describe all the phenomena completely accurately (not just in a perturbative approximation) even though the theory may also endorse the picture in which there are many wormholes.

The reason is that the spacetime topology isn't a well-defined quantum number. It is not associated with any well-defined linear operators on the Hilbert space! Some particular ket vectors in the Hilbert space may be "suggestive" of one spacetime topology but their linear superpositions may prefer a completely different topology!

So string theory works perfectly even if you try to "impose" an arbitrary spacetime geometry – and topology – on it. It may just make a particular process look more contrived than it is in a different description. Particular configurations may be closer to the "ground state" of one spacetime background and the corresponding interpretations may be pragmatically preferred. Some spacetime backgrounds may be preferred over others (in which a physics problem may be described equivalently) because they have lower values of the coupling constants or longer radii (or smaller curvatures). But fundamentally speaking, none of them is "qualitatively" better than others.

I've been sure about the string/M-theory's consistency – up to arbitrarily short distance scales – in a flat background (a theory seemingly "prohibiting" the different topologies) since the 1996 discovery of Matrix theory by Banks, Fischler, Shenker, Susskind. Why did Matrix theory settle these viewpoints of mine?

Well, in the early 1980s, Green and Schwarz were using the light-cone gauge when they were igniting the First Superstring Revolution. The light-cone gauge apparently allows you to slice the spacetime into null slices\[

x^- = {\rm const}

\] and all the interactions of strings may be described in this way. A good feature of a slicing is that it allows you to define a Hamiltonian. If you study string theory covariantly, i.e. you only ask about quantities that make the Lorentz symmetry manifest, you may see that string theory only calculates on-shell quantities (the scattering amplitudes in the S-matrix). The off-shell Green's functions (defining finite-delay response functions etc.) aren't well-defined. This has been known to be OK – in fact, it is a property exemplifying string theory's ability to teach us basic things about gravity – because the diffeomorphism symmetry makes the location of a field operator gauge-dependent so gauge-invariant (or gauge-covariant) correlators can't be associated with "fixed" spacetime points. The S-matrix formalism only gives us the unitary evolution operator \(S\) from \(t=-\infty\) to \(t=+\infty\).

But the light-cone gauge seems to overcome this limitation. It seems to give us a well-defined Hamiltonian \(H=P^-\) (using the light-like slicing) which may be exponentiated to produce the unitary operator for any finite or infinite delay, \(U=\exp(-iH\cdot \Delta t)\). The light-cone gauge makes the Lorentz symmetry a bit harder to prove and the diffeomorphism symmetry (and analogously, other gauge symmetries) are being obscured (they're not a part of this formalism at all, to be more accurate, because the light-cone gauge eliminates all the unphysical redundant degrees of freedom from scratch). But if you care about the finite-time evolution and the manifest absence of bad ghosts (negative-norm states), the light-cone gauge seems to be a way to go.

However, in the 1980s, it was conceivable that the light-cone gauge was only consistent in the perturbative approximation – and only for processes for which the initial spacetime background (typically the Minkowski flat spacetime) is perturbed at most infinitesimally. Can the same light-cone gauge describe processes in which the spacetime is heavily curved (by many gravitons) and for which \(g=\O(1)\) which also means \(\ell_{\rm string}\sim \ell_{\rm Planck}\)?

Matrix theory – and matrix string theory – answered these questions in the affirmative. The gauge works perfectly at any \(g\) – matrix string theory is a complete, non-perturbative definition of type IIA string theory at any value of \(g\) – and there is clearly no limitation on the number of gravitons you may add to the state, either. So at least the light-cone gauge (and perhaps, but not necessarily, other gauges) is a consistent description of a theory of quantum gravity in a nearly flat spacetime background, even at an arbitrarily strong coupling and an arbitrary energy density.

All these developments support the general theme that string/M-theory gives us the freedom to "impose" any spacetime background we wish and there will be a consistent description of the phenomena, anyway. In fact, it seems that you may even start with a background violating some laws (e.g. classical equations) and the stringy-braney matter will do its job to fix the bugs and list all the possibilities for configurations that are actually kosher (and their evolution).

In the early 1990s, this theme was also supported by the discovery of the topology-changing transitions in string theory that are completely smooth and consistent. One Calabi-Yau manifold was shown to be able to change to a topologically distinct one by going through a geometrically singular but physically well-defined and fully consistent intermediate point (a conifold). The milder flop transitions (which preserve the Hodge numbers but rearrange the cycles into their combinations) and the more severe conifold transitions (which do change the Hodge numbers and connect all or almost all Calabi-Yau manifolds into a connected web) may also be described as a condensation of wrapped D-branes on the shrunk cycles (or processes occurring on them have extra world sheet instantons wrapped on the shrunk cycles). In this sense, any state "on top of" the Calabi-Yau manifold of the second topology may be described as a state (including wrapped D-branes) on the Calabi-Yau manifold of the first topology and vice versa.

Again, this freedom to think about the topology of the Calabi-Yau background agrees with the general lesson that string/M-theory gives us the freedom to "choose" the spacetime background without losing an iota of the consistency and accuracy of the description of the resulting physical phenomena.

Maldacena and Susskind: don't be afraid to start with a vastly different spacetime

This general point is strengthened by the recently articulated ER-EPR correspondence by Maldacena and Susskind. They phrase the insight as an alternative description of the entanglement (popularly associated with the acronym "EPR" of physicists who popularized it and demonized it) and with a non-traversable wormhole (the Einstein-Rosen bridge).

In this description, one is dividing physical states to "spooky ones" and "ordinary ones" which introduces a clear asymmetry to our emotions. The unentangled black holes are the "ordinary ones" while a wormhole is spooky; and any general entanglement is also spooky. This is a popular way to present the correspondence – especially because the readers, and not only the complete laymen, have been brainwashed by decades of books painting entanglement as a "spooky thing". And the wormholes (even the non-traversable ones which are almost certainly allowed) have been largely underdebated in the literature. So the ER-EPR link identifies two spooky or bizarre things with one another.

If you subtract this low-brow laymen's bias and realize that there is nothing spooky about the entanglement (or about different topologies), you may formulate the ER-EPR correspondence in a more symmetric and more rational (my) way. It says that
The Hilbert space \(\HH_{ER}\) of the states of an Einstein-Rosen bridge, including all possible excitations of matter in the bridge, is the same Hilbert space as the Hilbert space of the two disconnected black holes at the same locations, \(\HH_{BH1}\otimes \HH_{BH2}\). These are just two ways to interpret the very same quantum states in the same theory, two ways to associate them with different spacetime backgrounds.
This meaningful beef of the proposition above is exactly equivalent to the correspondence as formulated by Maldacena and Susskind. But the two Hilbert spaces that are said to be equivalent are treated much more symmetrically. The two Hilbert spaces are the same for one bridge or a pair of two black holes. Just the states of the two black holes that have a lower entanglement entropy may be presented more meaningfully as states of two black holes; the higher-entangled states are more pragmatically described as states of the single Einstein-Rosen bridge.

Equivalently, the states with a nearly empty Einstein-Rosen bridge are more meaningfully described as not too violent excitations of an empty Einstein-Rosen bridge; the strongly excited Einstein-Rosen bridges correspond to bridges that tend to "chop off" and resemble the two independent black holes.

You see that the ER-EPR correspondence is the latest salvo in the same theme: string/M-theory allows you to choose many different spacetime backgrounds if you want to describe the same physical phenomena. But there's a sense in which the ER-EPR correspondence goes beyond all the previous examples (coherent states of gravitons, T-duality, mirror symmetry, heterotic-K3 duality, AdS/CFT correspondence, conifold transitions). What is it?

In all the previous examples that showed that the character of the spacetime could have been modified, the time didn't really play any substantial role. The two (or many) spacetime backgrounds whose physics was claimed to be equivalent were actually using the same spacetime coordinate (or at most a mildly curved one, in the case of the coherent state of gravitons). The bulk of all these dualities and similar relationships could have been described in stationary situations.

But in the ER-EPR correspondence, time plays an essential role for the first time. There is no natural way to choose the time coordinate in a way that would apply to both equivalent descriptions (two isolated black holes; one bridge). Despite the fact that the equivalent description changes the shape and topology of the spacetime in a way that can't be reduced just to a change of the shape and topology of the space, the two descriptions are still perfectly equivalent.

This seems to imply that the lessons about the "allowed topology change" in string theory that we have learned from the previous examples does generalize to the case of the spacetime (and time), too. The chronology of events (or knowledge about observables) must still be preserved and all the valid equivalent descriptions will agree about them. But the precise way to connect them and embed them into a spacetime doesn't have to be unique. Whatever way you choose, string/M-theory will offer you a valid description of the physical phenomena. You should never be afraid of choosing a background. String theory will work independently of the background and give an interpretation for the phenomena.

So are the space, time, and spacetime doomed?

I think that the answer is the following: Although it could have happened that quantum gravity could have banned the usage of spacetime geometries in the accurate treatments of situations and processes (in the Planckian realm), the actual evolution of quantum gravity and string/M-theory seems to paint a different story.
We are actually allowed to use a spacetime background, even if you want to describe or predict the most generic or extreme processes absolutely accurately. In fact, we may use (almost) any spacetime background to describe (almost) any physical processes.
Allowed topology changes (e.g. conifold transitions) may always be described as a condensation of objects on the original background; allowed (non-traversable) wormholes may be described as entangled states of objects (on a wormhole-free background) whose details reflect the properties of the wormhole.

So what is doomed isn't a spacetime in the sense of our right of using the concept; what is doomed is the idea that a classical spacetime is unambiguously associated with states or processes. The association is not only refusing to be unique but to a large extent, the spacetime may be chosen arbitrarily (even its topology may be chosen differently, even if the topology change involves time in a nontrivial way). An inconvenient choice of the spacetime background for a given physical situation will be manifested in the complexity of the collections of particles and objects that live on top of the chosen spacetime but the theory will never say that the choice is "fundamentally wrong".

So the advances in string/M-theory and quantum gravity have eliminated the idea that the reality canonically carries a "particular spacetime background". At the fundamental level, the separation of the reality to the "spacetime" and "the objects that live on the spacetime" is highly redundant if not arbitrary. This fact may be viewed as a manifestation of the inevitable unification of gravity with other forces or matter. You just can't objectively or canonically determine which part of the state or process or reality is gravity (space and time); and which part is the information about the objects or phenomena taking place on that background! Only some combination of theirs retains its unambiguous meaning.

The relationship with the amplituhedron was already discussed: Similar constructions may make the spacetime look totally obscured but the ultimate reason why the amplituhedron is more important than other generic polytopes is that it has implications for phenomena in a spacetime – relationships between knowledge of observables associated with spacetime points or regions – in fact, phenomena in many spacetimes. A spacetime will always be able to pump life into mathematical results (although it may be done in many ways). Without any spacetime, claims about mathematical structures will always remain pure maths or – using a physicist's perspective – unphysical tautologies.


  1. Hi Lubos,

    Don't get discouraged to write about cutting-edge physics!

    I'm an algebraic geometer with a strong interest in physics. I systematically enjoy your physics posts, and learnt an awful lot from them!

    I'm not sure whether I'd pass all your filters or not, but I can assure you that your physics blogging has had an impact on my understanding of physics and on the way I think about my research.

    So thanks a lot for it! :)

  2. "Does it make any sense to write about advanced scientific topics?"

    Yes, dear Lubos, it does. Yours is the only site I know that discusses these issues in essay form and, while I may not know 95% of the math you use, there is still a kind cumulative intellectual appreciation, I feel, to humming through to the end of every post you write, which I try to do. Where else can we eavesdrop on real "physics talk," or something close to it? I consider it a real treat and a privilege to learn how much I don't know about the beautiful language in which nature is written (and one of the minds that can read it). Your stuff is forever.

  3. On the subject of "grandma's summer" ;-)


  4. Dear Lubos,

    If you were to quit writing about cutting-edge physics my quality of life would be diminished. It is selfish of me to expect your free contributions to my intellectual stimulation but the fact is that you have made my retirement a great deal more interesting.
    To lose TRF altogether would be even worse.

  5. Re. the magnet in a copper pipe, magnetic damping has been used in practical applications for more than a century. Two common examples are precision balances and electrical meters.

  6. please, please, please, don't stop! WE ARE following you...

  7. Yes, one of my favorite songs from Mladek. Lucretius might know Mladek because he is quite popular in Poland as far as I know, at least his "Jozin z bazin" song
    My assotiation to a song about autumn is this song from Moody Blues

  8. Thanks, Kashyap, the flat space is only easier to the extent to which it's empty. I am just saying a trivial thing that you may view all topologically trivial but curved space - and probably even different-topology space - as the flat space with some particles (coherent states of gravitons, entangled black holes, wrapped branes etc.) added on top of it. To get the equivalent state as a state built upon a flat background, you must add much more stuff than if you start with a more optimized curved space. So it is *not* easier to work with the flat space, quite on the contrary. It's easier to work with a mildly excited curved space than massively excited flat space.

  9. LOL, good that we find at least some overlap with music. BTW when it comes to the autumn and Mládek, I shouldn't have forgotten "The most beautiful season is apparently the autumn". I can't find the original sung version on YouTube, just this non-original instrumental one:


  10. I too, over the years, have been using Lubos' posts to teach my son physics. His clear, simple, sometimes novel, but always correct expostions are only equalled by Feynman's ( whose output was far less). Though more than happy to take advantage of it, I envy his ability to write, in what seems to be a matter of hours, what would take me days or weeks. TRF is almost indispensible in our house.

  11. Wow!
    'Ask and ye shall receive'
    It's twue ! It's twue!
    Thanks Lubos!

    I'm also very pleased to see so many comments from 'satisfied customers'.

  12. oh, I just wrote that comment there but I forgot to say that I am too a "satisfied customer" ... :)

  13. Hi Lubos. I don't know what the best way to contact you is so I'll just comment on your most recent blog post.

    I don't know if you're aware of the insanely detailed Wikipedia article on loop quantum gravity: https://en.wikipedia.org/wiki/Loop_quantum_gravity . It is in part written by Carlo Rivelli, but most edits are from somebody named Ibain. It is much longer and more technically detailed than the article on string theory, and I'd guess any topic in actively researched physics (it is longer than the article on quantum mechanics and about the same as GR).

    One could give the benefit of the doubt and say that they are just trying to provide information, or take a more sinister view and see it as a continuing strategy of swaying public opinion about quantum gravity, not being able to do so through proper scientific discourse.

  14. Lubos - The latter sections above seem be to making the point I was laboring towards in the comments to your earlier post:
    http://motls.blogspot.co.uk/2013/06/maldacena-susskind-any-entanglement-is.html .
    It seems to me that the equivalent descriptions using different backrounds correspond to different tensor decompositions of the total Hilbert space.
    Am I wrong about this? If so, what stupid mistake have I made?
    If I'm right, I have more questions--
    1) How does this square with your statement, in the comments mentioned above, that all inertial observers share a common tensor decomposition due to locality?
    2) Could it be that the observables of an infalling obsever and a stationary external observer have different tensor decompositions? More precisely - does observable O in, say, the stationary frame become observable O', with a different
    tensor decomposition, in the infalling frame? The point being that the stationay observer infers a firewall while the infalling observer moves peacefully through the horizon observing no such thing?


  15. Is it true or anything like true that New England has the best autumn foliage in the world, or is that just some baloney that someone made up? (As a kid in New England, I took it as an article of faith.)

  16. LOL, right, but I am not afraid that too many people in the public open the Wikipedia page on loop quantum gravity.

    Loop quantum gravity is only used as a non-string theory in various media tirades but no one really gives a damn about that set of ideas.

    It's appropriate for such an amateurish piece of pseudoscience that it's being built on Wikipedia and not in the usual expert research channels and I guess that sensible readers of that Wiki page will be able to see that nothing exists beyond the demo on the Wikipedia.

  17. Dear RAF, absolutely. An isolation of a background for a given subset of the Hilbert space or subset of processes does define a locality and the corresponding tensor decompositions of the degrees of freedom.

    1) It squares perfectly - well, there is no contradiction simply because all the constructions relating the different backgrounds in this blog post are non-inertial, at least one side of the relationship is. The absolute character of the tensor decompositions is invariant (due to locality) under the Lorentz transformations that don't really change the invariant properties of the background geometry but in this whole article, we're discussing exclusively more general transformations that *do* change the background geometry!

    2) Yes, the tensor decompositions of the two observers differ at least in some respects - after all, it's only the infalling one who sees the black hole interior operators.

    But it is nonsensical to ask whether the stationary observer infers a firewall. A firewall is/was a hypothesized property of the infalling observer's observations. The event horizon is like the Iron Curtain (imagine Berlin Wall) and the firewall is the shooting at the Soviet bloc citizens who try to get through the Iron Curtain. The question whether this firewall exists can't be answered just by having obedient comrades who live inside DDR (the stationary observers). ;-) The point of AMPS is that the firewall has to exist for the infalling observer. Of course that the stationary observer won't be able to see/deduce/infer a firewall in any way.

  18. LOL, I've heard this comment about New England many times. But frankly speaking, I have also heard the same comment about China.

    I am not empirically able to see how the New England autumn colors are better than the Czech ones. But some people offered some biological explanation, something about the shortage of nutrients in the soil or something like that.

  19. The event horizon is like the Iron Curtain (imagine Berlin Wall) and the firewall is the shooting at the Soviet bloc citizens who try to get through the Iron Curtain.

    LOL, did you come up with that simile just now?

    Funny how that wall-of-fire idea (I'm being nitpicky; a "firewall" is what stops a fire and keeps people safe) inspires people.

    Very nice and accessible write-up by Prof. Karch, by the way. But my favorite is still Dennis Overbye's NYT article:

    Because of this wormhole connection, Dr. Maldacena explained, “Ted and Bob are the same.” So the result is sort of like the happy ending of one of those screwball romantic comedies that involve mistaken identity and the handsome vagabond turns out to be the prince in disguise; Alice can marry Ted who is really Bob and the bonds of matrimony extend smoothly across the edge of the black hole.

    Genius ;)

  20. This post proves once again that Luboš is one of the top thinkers today. What a clear & sharp exposition! Thank you for writing this up and it has cleared up a lot of things for me.

    It's a shame that Carroll gets so much press these days when the guy is clearly confused about so many things.

    PS: "Off-topic: Does this really happen to a neodymium magnet in FAT copper pipe?" Yes it does and the pipe doesn't even have to be thick. You can drop a small neodymium magnet down a thin copper pipe (those used for plumbing) and it will fall down slowly. In a 1ft pipe it will fall for about a second. As it falls down, it induces current in the pipe that creates magnetic fields that will be pushing the magnet in the opposite direction that the magnet is falling due to gravity.

  21. LOL, thanks, I just went to mindlessly write an analogy with borders of countries and started with the U.S.-Canadian border. Then in 5 seconds, I recalled how incredibly boring and non-firewall-like the border was (I was only crossing it for a few meters at the Niagara Falls and the 1,000 Islands) so I decided there has to be a more firewally border and it was closer to my home, of course. ;-)

    Overbye is a fun writer and he sort of knows what's shaking among the top experts, too.

  22. Thanks for your excessive compliment and the insightful explanation - I see, so the influence between the objects in both directions is important for the experiment. Cool.

  23. Yep! It even works in aluminum tube (any non-magnetic metal will do really). I believe it's due to Lenz's Law.

  24. Eelco HoogendoornOct 23, 2013, 10:42:00 AM

    We all know the jokes of the form 'how many physicists does it take to change a lightbulb'... but youd think the answer to the question 'how many physicists does it take to appreciate the most basic implications of electromagnetism' would be 'any single one of them should do'.

  25. LOL, you're probably right. I should have thought of that.

    But you may be underestimating electromagnetism and its consequences. It is a theory about the set of phenomena that include all of chemistry, biology, condensed matter physics, and almost all of engineering, among many other things. From this perspective, it's a bit questionable whether a piece of metal in another piece of metal that looks "just like f*cking", using a Feynman's artist friend's words about a very similar physical setup, is the most important consequence of electromagnetism. ;-)

  26. Concerning the arrow of time. Are you sure that the thermodynamic arrow of time is the only explanation for time asymmetry? There is a time asymmetry on a more fundamental level of particle physics
    I am also not so sure if the termodynamic arrow of time is the right explanation why time seems to flow forwards and not backwards. In some open systems the entropy is increasing and not decreasing, does it mean that they have an opposite arrow of time to the rest of the universe? (i.e. a growing embryo in mother's womb is a system with decreasing entropy). Also, entropy changes have different speed in different systems, espetially in systems far away for terman equilibrium, does it mean that time flows with different speeds there? The termodynamic arrow of time just doesn't seem fundamental enough, but is emergent. I believe that is the reason why many people don't want the accept it as the fundamental explanation for the arrow of time.

  27. Thanks Lubos.
    It's nice to know that my initial intuition was correct.
    Of course you're quite right about the firewalls - I was up rather late and managed to confuse myself; I'll try not to let it happen again.
    Thanks for taking the time to answer.

  28. Yeah, the poor soil thing is part of the story I heard as a kid.

    BTW, the book "1491" (which I recommend) says that New England was not inhabited until about 2,000 BC because before then it had not sufficiently recovered from the glacial period.

    "1491" is about findings about the Americas and the Indians which have appeared in the past 30-40 years.

  29. Eelco HoogendoornOct 23, 2013, 12:11:00 PM

    True, it is nontrivial to see all the implications. On the one hand; how much simpler does it get; the exterior derivative of the two-form is zero; you cant come up with a simpler wave equation over a spacetime. On the other hand, understanding what all the parts are really up to can be quite tricky. I always struggle especially in trying to make the connection between classical electromagnetics, and trying to impose a (virtual) photon interpretation on top of that. Someone should write a good textbook about that.

  30. Eelco HoogendoornOct 23, 2013, 12:13:00 PM

    I would add to that; its not the 'most important' implication of electromagnetism; but it is one that follows relatively directly from the inductive term in the Maxwell equations.

  31. Just a trivial but nonetheless needless non-mathematical mistake spotted: "may still" should be 'may stay'.

  32. Holy shit what nonsense! If some fucking dumb-ass neuroscientist thinks mathematics and physics are just constructs of the human brain I invite him to get a lobotomy and see if suddenly 2 + 2 = 5 or he can now fly.

    This notion actually pisses me off and I would hope no other TRF readers are taken in by such crap.

  33. when you are interested in what nature really is, your understanding reaches sooner or later its limit on the technical level. there's too much to process. as lubos says, nature is by far more clever than us. but there were, are and will be exceptional thinkers, like maldacena to name one, with the ability to extract some gems from this mess. at this point, only few initiated (old) eyes can see it shine. but it seems that the new knowledge can always be purified (at least partially) from its technical gangue to be seen, at last, by many more, and especially the younger one. and i think that it is an important part. as lumo pointed out many times, old generations of scientists are often left behind by the younger ones, who learned the new knowledge as if it were always there. i am now (maybe after some TRF phase transition of a sort) more and more convinced that this purified knowledge can and must be taught, and the sooner the better.

    as a teacher, i am often disappointed to see these 18 years old humans unable to have a clear view of the atomic constituents, when my 5 years old son plays to built mesons and feeds W bosons with higgses. he lives in a 10 dimensional curved spacetime partially compactified by that funny tachyon, with wormholes connected to the djinn's universe where quarks can be separated (but you have nonetheless to pull very hard) and so on. by the way, i was delighted to see that he was already able to do a clear distinction between the observable universe and the djinn's world where everything goes, while some deluded mature crackpots are stucked forever in a mixture of fantasies and reallity.

    it's so easy to replace all kind of religious superstitions and to enlarge the imaginary playground of a child. you just have to pick ideas out of the boundless imagination of nature herself. but this knowledge has to be accessible in its purest and sexiest form, and lumo is really good at that.

    my last point, is that the process of purification can be very very slow, for example if your only source of information about nature little secrets is from the public school program (that was my case) or, completely distorted by dishonest cranks (sometimes also in the public school program, as i have seen recently). that's where the very existence of some natural catalyser like lubos' brain seems to be one of the blessed gifts that Nature provides in her infinite goodness.

  34. The SLAC study that you reference adds nothing new, Mephisto. If you were to use the phrase “statistical arrow of time” rather than “thermodynamic arrow of time” you would understand the obvious fact that SLAC study required a large number of events in order to get a statistically significant result and that is the whole point.
    Amateurs such as you fail to grasp the essential science and fall back on mere words, which do not carry the rigor needed to make scientific progress.
    Regarding entropy in open and closed systems you really have not been listening at all. Shame on you.
    I admire your willingness to ask questions but you really need to ponder what Lubos has written and discard the illusion that you might have something basic to contribute.

  35. Please read my above comment to you and drop the illusion that you have anything useful to say. It is OK to ask questions, Mephisto, but you are out of your league here.

  36. I suppose the correct answer to my question is Obvious if i were to read all the many excellent posts you have made on stringy QM, but I hope my ignorance is excused and you will point me correctly.
    I thought it intuitively obvious that if I am a string I must experience time in a different metric than if I am in space observing the string. The string must vibrate and that clock would be a time and thus proper to it. But from outside the string vibration must be a mass property. if not the string would be moving in our spacetime using it's clock. and that would mess up causality. but if it ack degree of freedom in time then from outside it's clock is instead mass energy - like knowing how tight clock wound by weighing it but not being allowed to see its face.. That way insteaad of e=m(1*1) e=m2 and c doesn't involve self and we conserve causality at scales where it matters. The string not convey information about time in the momentum to me also seems intuitive because dynamics is only meaningful to anything but the particle/string if it is relating but it cannot be even relating to itself if it has no clock A spacetime would not be a universal source unless it has a master clock which seems impossible and verybcounterintuitive at least. Similarity if anything is approaching a black hole it's internal clock must dilate to outsiders until stopping to outside observers when instead it can be perceived only as mass term of the whole hole. So crossing the horizon means losing that degree of freedom for time and it becomes mass.
    So if any of this were so there would be seemingly no reason to expect a fixed spacetime and causality would never be at issue?

  37. Edit not working meant e becomes mass squared not mass times two

  38. One of your best articles in this blog, Motls!

  39. Keep on Lubos! I have to say it. The Reference Frame is the reference frame for many of us disconnected of theoretical physics world, and a voice to listen when in doubt. Personally, when I hear something that switch on some bullshit-alarm, I always ask to myself: What does think Lubos of this thing? and then I go right away to Google and type:

    "(doubtful stuff) site:motls.blogspot.com"

    (And not to mention the hilarious reviews on climate hysteria, funny and educational. Just great.)

    So thanks Lubos for your awesome job, the fact that there are people who will not be able to see it, is a sign that something is wrong in this world... Luke 23:34: "Father, forgive them, for they do not know what they do". ;-D

  40. Gene, the whole point of asking questions is to learn something new. I like asking questions like what is time, why is the time dimension so fundamentally different from the spatial dimensions, but at the same time mixes with the spatial dimensions. Space exists at an infinity of points, but time is different because it seems to exist only at one point - the now. There is the past, the present and the future. Why is the present so special? From the special theory of relativity we know that space-tiime exists as a single block in one piece and the different observers slice this 4D space-time according to their coordinate frames. But it seems that the present moment is only an illusion of consciousness, because what is fundamentally real is this 4D eternally existing block of space-time. But if time is only an illusion and the 4D space-time block exists eternally, it means that everything is predetermined, even the outcomes of quantum probabilities. If these ideas seem strange that read Brian Green's book The Fabric of the Cosmos where he even has nice pictures of this slab of eternal space-time and its 3D slices. Why do we have the idea that time seems to flow when in fact time is an illusion?

  41. Nobody knows the answer to this question, Mephisto. Maybe some time in the future humans may know the answer. It is not the first time that philosophy turned into science. But the way science moved forward is to think about questions which at least in principle lead to observable consequences or theories which may in principle answer them. Try to convert your question into a physics question.

  42. Philosophy can inspire physics. Mach's principle inspired Einstein to create his GTR. The philosophical belief that physics should be the same for all observers led to STR. Bohr's principle of complementarity was also more a philosophical principle than physics. And analogously, if you have some philosophical insight about time, it might help you create some physical theory. Physics without philosophy is boring meaningless calculation.
    Time is philosophically one of the hardest problems. The only insight about time that physics was able to make is that eggs break and never unbreak. Thanks to Boltzmann we know why. But is it the whole story?

  43. Pondering the essence of time is very rewarding intellectually. But one can't reach the bottom of it while looking at it through "a fog". Thanks Memphisto for the inspiration! :-)

  44. Thanks again for the inspiration! Here you go... http://www.toebi.com/blog/theory-of-everything-by-illusion/time/

  45. If your lifestyle is causing you to spend more money than you should, you should try to live within your means and start saving. One of the main causes of bankruptcy is credit card debts. Credit cards enable you to spend beyond your means, and you will find yourself facing debt problems if you do not control your spending. When you are experiencing financial difficulty, you should only purchase things that you need and try to take advantage of any opportunity to save money. Make a point to save a certain amount of money every month, and use that money to reduce your debts.

  46. If only - that would be great : the border guards would think they shot someone, whereas the person himself could pass unharmed - but that would start to look like may worlds theory

  47. Wow this is such a nice review article about the many ways how string/M-theory does not depend on a particular (spacetime) background, and the unification of gravity with everything else aspect I already noted in the previous linked article to the topic, is very cool too :-).

    Of course some examples I do not understand in all detail, as the one involving wrapped D-brane condensation to go from one CY to another, but maybe the baby book I am currently reading (I dont say what :-P) will help me to at least roughly and qualitatively understand better how such things work too. It always happens to me that the more I manage to learn about STuff myself, the more I like and get excited about it :-)

    BTW I am not sure how much good being so much interested in and excited about fundamental physics and in particular reading TRF does to me ;-):
    It definitively change my way of thinking and looking at things (physics and maths that I learned previously is much clearer to me when a serious theoretical physicist explains them from a fundamental (conversely to purely applied or phenomenologica) point of view. And stuff that previously was interesting enough for me to do now seems way to "flat and shallow" or not exactly "rocket science" to excite me ...

    So dear Lumo, please keep it going, but not too much faster than I can read ... :-D


  48. Quantum Experiment Shows How Time ‘Emerges’ from Entanglement

    Time is an emergent phenomenon that is a side effect of quantum entanglement, say physicists. And they have the first experimental results to prove it


  49. Wow, I totally forgot to read this article since so many days!

    I finally read it and I'm like "YAY!" as I read the "T-Duality and other operators" section.

  50. One of the first things to do is to get some advice. There are a number of charities and organizations that can help you, such as the National Debt line or Citizens Advice Bureau. They will be able to guide you through your options.