So, yes, so today's lecture, well, I planned now not to be shy of using some formulas.
If you don't understand it, then I'll tell you when again to wake up, because then the
rest will be a little less technical again.
So we were, well, one of the earlier lectures, I was talking about the black hole.
And the black hole is going to be a very central issue in today's talk.
I claimed in that lecture, I think it was the second lecture, that there is a difficulty
with black holes.
If you try to put them in our theory of particles or quantized fields and such, something seems
to go wrong.
And what goes wrong is the information that went into black hole doesn't seem to reappear
properly.
Now, I have lengthy arguments about this, in particular in string theory, where the
partitions of string theory are telling me that, yes, they can, for such objects, they
look like black holes.
And yes, they do behave properly, so what are you worrying about?
Well, I'm worried about the fact that those objects are still not quite well understood.
And of course, the link from there to quantum field theory of elementary particles is a
very big step.
I want to see more directly how quantum field theory of elementary particles can be made
in agreement with the postulates we have for black holes.
And the claim is there's something interesting and non-trivial we will say about that.
So let me again go to Schwarzschild, but today I'll now mention some equations every
now and then.
So Karl Schwarzschild wrote down the exact equations, the formula for a solution of Einstein's
equation, and in modern terminology, it looks like this.
He wrote it slightly differently, and then he tried to argue away the difficulty if 2m
equals r.
If 2m equals r, then this denominator vanishes, and Schwarzschild thought that that was a
singularity.
It isn't.
It is just the horizon of a black hole.
The strange thing is that the same coefficient occurs here, so we see that the coefficient
in form of dt squared vanishes when you reach the center of the black hole at the distance
r equals 2m, and it just means that time stands still at that point.
That's very roughly the interpretation of this equation.
So we had this similarity, which seems to be rather innocent because it's fairly easy
to remove it by coordinate transformation.
Structures yourself didn't realize that, but this similarity is even stranger.
That tells you that time stands still, and if you go r less than 2m, somehow the sign
here switches, and this sign switches, so r and t switch role if you go to that point.
Very strange things, and it does not very well understood what it means.
So let me now again describe the space-time in black hole, but now in terms of a formation
that is slightly more convenient to describe, I have here what one could call an artist
impression of space-time.
Time runs vertically, space runs horizontally, and here you have imploding matter, and well,
the local light cones then indicate this way.
This means that if space and time are like this, far away from the black hole, the light
cone is such that rays going out go out in this direction, rays going in go out, go in
in this direction, rays going sideways come, of course, towards you or away from you.
They're all lying on this local cone, and now you can ask, oh yes, the orbit of a physical
particle has to go slower than a speed of light, and therefore it has to go inside the
cone wherever you can see how the cone goes.
So we can draw at various spots, we can draw those light cones, and then the situation
looks like this, that we have light cone oriented models in an ordinary direction
before the matter went in, but after the matter goes in, then the light cones get tilted,
so they are pointing inwards, so to speak, and you can actually draw a line, a surface
here that just touches the light cone, and that means that if you are on a boundary line
between the blue and the green, then you can try to escape, but you see locally, you have
to escape with the speed of light, and you're getting nowhere, because you're just moving
straight, you're not increasing the distance from the black hole anymore, and that is called
then the horizon.
So now we can draw a surface there, and say once you're inside that surface, you're lost,
because the local light cones are pointing inwards, and this dotted line here is inside
the surface is the singularity.
So now you see why I said the day before yesterday that that's beyond infinity, because
once you're on the light cone, it takes infinite amount of time for any signals to get out,
so an observer at late times doesn't see what happens here, but has to wait a very long
time for those signals to come out, and that observer never sees the signals that are produced
inside this surface, so that's how space-time rough looks for a black hole.
The radius can then be computed to be 2m over c squared.
I had in the previous expression just 2m, but then g and c squared just absorbed in
the symbol m, so that if you put these constants back in, then this is the expression.
This is the radius of the horizon of a black hole.
Now yes, we often divide the universe into region one, and region two, region two is
sort of hidden, region one is the outside.
This point S is a very important point, because all the Hawking particles seem to emerge
from this point S, and so this is a Hawking particle, so you see it gets detached from
the horizon very, very slowly, eventually it gets out, and then it's a radiated output,
but all these Hawking particles originate from the same spot, and so you might think
there must be a very heavy cloud of Hawking particles out here, because these things shining
these particles eternally, but strange enough, there's nothing happening here, so these
are just vacuum fluctuations, and vacuum fluctuates, but we don't see that, the vacuum around is
also fluctuates, and we don't see a thing of that, so a local observer sees nothing
of the sort, but it's due to this strange mix-up of space and time that a late observer
then suddenly sees particles coming out.
Now this was the idea of the black hole, you see other oscillations that occur also at
the same spot, just don't make it out, and they disappear back in, and this was the difficulty
that we have to consider when studying Hawking radiation.
Now I want to mention the concept of black hole complementarity, and that also was mentioned
briefly the day before yesterday.
This is one observer, in this case Matt Hatter, who went into the black hole and who sees
the particles going in, the other observer, in this case Alice, doesn't see the particles
going in, but sees the Hawking particles going out, and black hole complementarity states
that somehow these two pictures ought to be equivalent.
So both Alice and Matt Hatter here can give a complete description of what happens in
the black hole, but in order to achieve that, you have to modify some laws of physics.
Now this is actually a better way of writing the coordinates of a black hole.
Traxio did not know that, but Kruskar and Sekiras, actually two people, realized this,
that you can redefine the R and T coordinate by writing an X and Y coordinate instead,
but the product of X and Y is this function of R, and you don't have to remember all the
details here, just the product of X and Y is a function of I, the ratio of X and Y is
a function of T. Now if you use X and Y instead of R and T, then this is a rather singular
transformation, certainly when X and Y becomes small, so the X and Y coordinates are then
indicated in this picture here, that's the R coordinate and time, now time runs in this
direction, but the X and Y lines are the red and green lines here, so red and green form
a decent looking grid in this space time, but the R and T coordinates are very singular,
they blow up at this point, this is where the horizon is, so the horizon can be compared
to the North Pole of the Earth, our usual coordinates, longitude and latitude, become
singular on the North Pole of the Earth, but it doesn't mean any polar bear cares about
it, they're just all straight through, so the polar bears would use more regular coordinates
here, but we, since we never usually come close to the North Pole, we are happy with
longitude and latitude, so if you use these coordinates, you discover, rather surprisingly,
that the singularity at I equals to M has completely disappeared, I equals to M where
X and Y is zero, I, X zero, or Y is zero, then you see that, that means this expression
has to be zero, and that means I equals to M, so those are these black cross lines, that's
now where the horizon sits, and so this is, so this is the new expression for the metric,
and that expression is totally irregular, except when R goes to zero, that's the true
singularity of the black hole, where things do become singular, so this is somehow an
easier way to describe the black hole. Now, oh yes, here are the equal time lines, so you see that
those equal time lines are just like the longitude at the pole of the Earth, it becomes singular,
so this is the thing that you might want to remember, not all the details, but roughly this
is a non-singular description of the metric of a black hole, and X and Y are the non-singular coordinates.
Now, and now the black hole complementarity says that you have two ways of looking at a black hole,
one would be seen from a distance, the astronaut never gets through the horizon as seen by distant
observer, he just sticks on the horizon and never gets through, but the astronaut himself doesn't
agree and thinks he or she is going straight through the horizon, so complementarity tells you that somehow
this observer should both have a pretty complete picture of what happens near a black hole,
so an observer going in experiences the original vacuum, but sees no Hawking particles,
the observer staying outside, sees no objects behind the horizon, but does observe the Hawking particles.
The claim is that they both look at the same reality, this is a situation that you also encounter in ordinary
relativity theory, where you can consider a moving observer and an observer at rest, and they both
are able to make a complete description of what happens in space and time, but their descriptions
differ from each other, so you need a dictionary to go from one picture to the other picture,
so that dictionary is now called black hole complementarity, and now what I want to do is
I want to explain my own theory about this, which is not yet very much accepted by many people,
it's just a wild idea, but I think it indicates that there is a way to use this to discover
something new, and that is the picture that we consider a large black hole, which is slowly
observing matter, but also slowly decaying, because it emits Hawking particles, so it does
both, if it does both, then its mass stays roughly constant, so it absorbs particles, it emits particles,
so you'd say its mass stays roughly the same, well but now let's switch on black hole complementarity,
I have an observer B, and B stands for black hole, and I don't know what this means,
things start to vibrate anyway, so observer B, who sees the black hole, sees the particles going in,
but in its metric original structure coordinates, those particles are never really going in, they are
slowing down while approaching the horizon, observer W, who actually looks at the white holes,
observer W sees only the particles coming out, so that observer works, oh yes, if observer B
doesn't see any Hawking particles coming out, but he sees the particles coming in, that observer B
would think that the mass is slowly increasing, because the black hole becomes heavier and heavier
according to observer B, now observer W sees only the Hawking particles coming out, and so observer W
thinks that the mass of the black hole becomes smaller and smaller and smaller, this is usually
not argued this way, so this isn't a way of phrasing the argument, it is not yet commonly accepted,
because you would also say well the real black hole just stays constant, it absorbs as much as it emits,
and that is it, that's the answer, but if you look from different coordinates, if you only
receive particles coming out and no particles coming in, then the law of conservation of mass
and energy would tell that the black hole itself should decrease in size, so the W says the white
hole, this observer sees a white hole, sees things coming out with nothing coming in, so according to
that observer the black hole loses mass, so that observer says that M1 is bigger than M2, and now
the question is who's right, there's something very crazy going on, and normally people say well
they're both not quite right, because the true black hole is not losing or gaining any mass,
because it's more or less an equilibrium let's say, but
I don't know what this thing is trying to say, so anyway we can
I go about this at length, but I think that it is relevant to consider the possibility that one
observer notices a different mass than another observer, so the consequence, now you can actually
accommodate for this situation if you introduce a new symmetry, and that is where I call conformal
symmetry, so suppose that the two observers disagree about the radio coordinate and the
mass coordinate, so the mass according to one observer is some function of t times a mass
considered by another observer, and the radius r as seen by one observer is the same function
lambda of t times the radius rho as seen by another observer, if you rescale things this way
you find that actually r over m remains constant, so this exponent stays the same for both observers,
and here you just get a rho squared, you discovered actually this Cascale metric, if you use the
Cascale coordinates which are light cone coordinates, and suppose both observers agree
about the light cone coordinates, then the only difference between two observers is a coefficient
in the front, so one observer, both observers agree about the black hole metric, but one observer says
this mass increases, the other says the mass decreases, and this agreement is just a coefficient
in front of it, a coefficient in front of the entire metric, so the nice thing about the light
cone coordinates is that the light cones are agreed about by all observers, and this was something I
wanted to put in, so I asked myself the question how can you accommodate a black hole complementarity
without violating principles such as causality in nature, suppose that the two observers agree
about causality, well that means they should agree about how the light rays are going,
because light rays distinguish spacelike separated from timelike separated, and those
signals can go faster than light, either according to one observer or according to the other observer,
so let's keep the light cones the same, so I want that the observers both agree about this part of
the metric, but they might disagree about the coefficient in front, that coefficient in front
does not affect the light cones, and that was the fundamental input idea, and now come
some mathematical statement, and perhaps here for some people the lecture might become a bit
technical, so we write down the metric of spacetime, which is usually given with this symbol, and they
say that one observer disagrees with the other by having a coefficient in front, and that coefficient
in front are called omega, or omega squared actually, and now you can go continue in different
ways, one would be to say that both observers split the metric terms of g mu nu in terms of
apart g hat mu nu and omega, where g hat could be defined just by saying that I restricted the
turbulence of g hat to be minus one, this g mu nu is a matrix, if I fix the term to be minus one,
then omega is what's left behind, and since this is a 4x4 matrix, and I've created omega squared
here, altogether the determinant would go like omega to the eighth power, so if I take the one
eighth power of omega, I get the determinant of the original metric g mu nu, so whatever that
metric is, whatever the determinant is, I define omega to be this, and then g hat is that, and then
g hat describes all the light cones, and omega just describes scales, omega just tells me how
fast clocks are going, and how large my yardstick is, but so the two observers disagree about clocks
and about yardsticks, but they do agree about causality and about light cones, and that was
the sort of input idea, so far away from the black hole, we also have this coefficient omega,
but there the metric is basically flat, and metric flat means a g mu nu is a trivial matrix,
or the identity matrix, and being identity matrix, omega isn't trivially equal to one,
nothing happens far away from the black hole, and moreover, if I were to black holes,
omega does have a meaning, yardsticks and clocks mean something, we are sitting in this room,
and you have watches that all run the same way, so we are using a time which is the same for
everybody in the room, we are using yardsticks that are the same, and all atoms in our bodies
have different sizes, and so on, so you can't scale up this room by factor two, and all the
people in it by factor two, you could say, well, nothing would seem to change, perhaps so, but the
laws of nature are certainly very, very much dependent on that, so the laws of nature are not
conformally invariant, but now we can say, well, that just means that far away from us, we assume
space time to be flat, in that case, this omega is fixed, it has to be chosen to be one, a different
way of saying that is that you can say that the vacuum state breaks the symmetry, so we call the
symmetry conformal symmetry, because it leaves light current invariant, if space and time have
been Euclidean and not Minkowski space time, then all circles and spheres would transform to circles
and spheres, and all shapes would locally remain the same, only the sizes would change, and that's
because the shapes remain the same, it's called a conformal symmetry, but this is called a local
form of symmetry, but the empty space time is not invariant under that, our space time that we live
in is very, very close to the empty space time, and there somehow this symmetry is broken, so one way
of getting out of the trouble that we say, well, all our atoms have the same weight, the same mass,
the same size, and the frequencies and spectral lines are all at the same spot, one way of getting
out of that is to say that the vacuum state breaks the symmetry locally, and this is just like the
Higgs mechanism, so what happens to conform a symmetry is just the same thing as it happens with
gauge symmetries when you switch on a Higgs field, so in a sense this omega is the Higgs field for
gravity, and so going from there, things really seem to become rather interesting, and that's why
I've tried to study this a bit more, so the idea is an omega can perhaps not be directly observed,
we do observe atoms, but only atoms are compared to the omega, assuming the omega is just equal to 1,
but if I would scale it, I would also have to scale all the atoms, so now the theory of black holes,
and this is very important, will only work if nevertheless scale invariance is an exact symmetry,
so the vacuum, the vacuum breaks conform a symmetry, but that's only if I exist in describing
the vacuum, I could actually choose only to do something else, but then I'm no longer describing
empty space-time with space-time with particles and matter in it, well that is the case near the
black hole horizon, all these observers when they differ from each other, one observer sees lots of
particles coming out, the other observer sees lots of particles coming in, some observers see nothing
of the sort, so all these observers disagree about how many particles are floating around,
and if one of them would say there's a vacuum, well then he has made one particular choice for
his omega variable, and that means that the vacuum actually fixes the conformal symmetry,
so this is the island that you have to use, and so the vacuum state and state of metric
are now both going to play a very central role in this theory, I have something to say about
the cosmological concept, but I'll leave that to the end of the talk, because I think there's
some very interesting things to be said about that, so now here comes again a little bit of
mathematics, so I'm sorry for those for whom this is unfamiliar, if you make a conformal transformation
of automatic of this in this particular way, then you discover that Einstein's tensor doesn't stay
the same, you can imagine this is a big modification, if I change the scales of things and the times I
think and have them change dependent on space and time, that's a very crazy transformation in
ordinary physics, and as a consequence the Einstein tensor changes completely, and when the Einstein
tensor changes, so must be the energy and momentum tensor of matter, TVU is a familiar
expression of physics that indicates how much energy, how much momentum density, and how much
tension there is in space and time, so this acts as a point in space-time, this is the energy,
momentum and stress tensor, this energy momentum stress tensor doesn't stay the same under a
conformal transformation, but gets modified and the first most significant terms are these,
they're less significant terms following in this equation which I didn't write down,
these are the most significant terms and you find that in general they don't vanish unless
lambda obeys a certain equation, if you insist that lambda obeys the equation where this vanishes,
then TVU might stay the same, well actually if I had a space-time where TVU vanishes,
then there is a non-trivial solution to these equations and actually if lambda would be a linear
function of x but not quadratic or anything else, just linear, then since this basically contains
second derivatives, this term essentially vanish, well essentially vanish if you do this in flat
space-time, well that means that the lambda could be linear in x, what does that mean? That is a
global conformal transformation, actually it's a transformation that mathematicians are fairly
familiar with, you can go from x to 1 over x, that is a crazy transformation but such
laws for instance like Maxwell's equations are invariant under that, even though space and time
look totally different under such a transformation, some equations in nature are invariant, so
that's a very interesting subgroup of the conformal transformations which are called
the global conformal transformations but local conformal transformations, I love the issues
anything, don't leave the TVU invariant, so if you now say I'm looking at the vacuum,
that's fine, you look at the situation of TVU vanishes but that fixes practically completely
your lambda, so that's how the vacuum breaks local conformal symmetry but not completely
the global one or a global one in a more indirect way. Now this is one of the important
transparencies of my argument which make the whole theory become interesting and that is
in engaged theory such as a standard model, where you have the Higgs mechanism going on,
yes or no, in some cases there's a Higgs mechanism, in some cases there's no Higgs mechanism,
I explained yesterday that for the weak interactions there's a Higgs mechanism,
for the strong interactions there's no Higgs mechanism, but in both cases you have a gauge
theory and the gauge theory means that you have to fix your gauge condition, that you know about
electromagnetism, the definition of voltage tension has to be defined at every spot in
space-time, then you know what the magnetic fields are and things like this, so that's called fixing
the gauge in electromagnetism. In this theory we have to fix the omega also by fixing the gauge
and fixing the gauge can be done in several ways, one could be to fix the component of the energy
momentum tensor, you could say I fixed the gauge by choosing omega equal to one, you can also say
I fixed the gauge by taking one of the components of the energy momentum tensor, the thing is 10
components, we choose one of them to fix the gauge and now I choose t minus minus, where
t minus minus really means that I'm using radial light concordance, much like Kruskal coordinates,
so minus minus means that for one of the Kruskal coordinates the energy and momentum
directed in that direction vanishes, this would be the outcoming energy and momentum, so if I exist
that no energy and no momentum comes out, then I'm looking at the observer who looks at the black
hole, he always sees things going in, he sees nothing coming out, so that observer without
really realizing it so far is actually used in this gauge constraint, no matter is going out of
the black hole, but you could do the opposite, you could say no matter goes into the black hole, then
you use this gauge condition, now these and these are not identical, well for ordinary world yes
because the amount of energy momentum in the world is very very small compared to what happens when
you have strong gravitational fields, so in general this is very close to that, but near a
black hole these two gauge constraints are very different, now this is beautiful because this
tells you that going from one formulation to another formulation is just a gauge transformation,
it is a local transformation to go from this to that, a local transformation tells you that
now you can map a black hole onto a white hole and this was precisely what you needed to get
the information back, because when you had only a black hole information within and nothing came out
that was wrong, there was something missing in that picture, but now we can do the opposite,
we can now transform from here to here and now we see that the black hole actually is a white hole,
information does come out, but you just have to make this gauge transformation and that could
rescue the information problem of black holes, so this is why I think this is so important, so
yes you can just read underneath here,
teaming has 10 components but you can only fix one of them and that by fixing one of them you fix
the value of this conform factor omega and that should in principle be enough to map
white holes into black holes and vice versa and this gives you a restoration of the information
problem, actually this mapping is what in particle physics would be called the scattering matrix,
the scattering matrix tells you that when you have a scattering process you first start and
only particles going in and you write down the exact state of the initial state of the scattering
process, think of LHC, there's either two lead nuclei or two protons coming in, that's the
initial state, then something very complicated happens physically and then particles come out
and that's the final state and in particle theory we calculate how the final state depends on the
initial state and the calculation is written in terms of a matrix called the scattering matrix,
if you can calculate the scattering matrix then you can calculate what happens when
particles collide and how many particles come out, so that's just the rule of the game in particle
physics, well it's the same rule of the game for black holes, there should be a scattering matrix
saying that you have employee matter making a black hole, something very complicated happens
and then matter goes out, that's very complicated thing that happens is expressed in terms of a
scattering matrix, this time the entire black hole itself is a scattering matrix and the way to
compute the scattering matrix is you have to make the gauge transformation from here to there
and so we go from the in states to the out states, this is the central theme of what I wanted to
say and but for this procedure to work it is essential that local conformance symmetry is
an exact symmetry and this is something new, people have gotten familiar with the idea that
conformance symmetry cannot be an exact symmetry because well as I said atoms have definite size
and spectral lines and so on so space and time or distance scales and time scales are fixed
but now in this theory yes they are fixed but only by the vacuum, if you think you are surrounded
by vacuum then you have fixed your space and time scales but not if you're very close to a
black hole with lots of lots of matter around, then you have ways to transform one picture to
the other picture, now the reason why physicists were not worried about conformance symmetry is
that conformance symmetry is in the use of these it is broken, explicitly broken and if normal
we call is an anomaly, now anomalies have always been very interesting in physics, we remember
exciting times when the standard model was born that the earliest version of the standard model
had also anomalies in it and you could ask what's wrong with that, for instance if you choose a
number of quarks different for a number of leptons then you have an anomaly in the theory, some things
go wrong and the answer was you must have the number of quarks equal, apart from effect the three
of color the number of quarks species must be equal to the number of lepton species, then the
anomalies cancel out, that is one particular anomaly the standard model, these anomalies are very
similar mathematically they take a very similar shape but people have got the use of the other
conformance symmetries are always associated with anomalies and well this time since that wasn't a
gauge theory and the ordinary theory is nobody cared about those anomalies, they are just
observed they are studied because they're interesting but they're not causing any harm
it was thought, now I claim yes these anomalies do cause harm because if you have a theory with
conform anomalies it that theory will explicitly break the concept that that black holes should
turn to white holes after a proper transformation, so we must somehow get these anomalies removed
from the theory and this was a very hard question how to do it and so we had to study conformal
gravity, so this transformation actually is a transformation you can also perform in ordinary
quantum field theories, in quantum field theories we can ask the question what happens if you
if you deform space and time in which these particles are moving and the result of that is
yes the theory looks some more or less the same for instance the light columns remain the same
so that massless particles move exactly the same way in these two theories but as soon as
particles have a mass then there's an effect, so now here again I'm doing some calculation
I'm taking the Einstein theory of gravity but usually in terms of the Einstein Hilbert action
so gravity is is often expressed in terms of a very compact notation saying that there's a
single single expression for the Einstein Hilbert action just the scalar curvature of space-time
that's it that's the action of of the Einstein theory and that generates the full Einstein
equations and everything it also generates the quantum theory if the quantum theory didn't have
such nasty divergences in it, well let's leave the divergence for a moment as I just consider
the combined action of Einstein theory and the action of matter but now split up the
matter tensor in terms of the g hat component and the omega because now this omega is going
to play a very special role what is so peculiar about this is that the omega dependent part of
the action turns out to be not unfamiliar in particle theory at all as I explained for a moment
but first a little bit of jargon of arguments without doing the calculation if you don't do
the calculation you would think that the miracle is happening what's the miracle well let's first
indicate over omega and leave leave g hat for a moment as it is just do the function integral
it's a double function integral one over omega one over g hat let's first do the omega integral
then the remainder is just an expression depending only on g hat well if you get an effective action
only depending on the g hat that action must be conformally invariant why well I integrated out
all the conformal invariants all the non-conformal invariants by integrating after omega I must
by construction must be left over with a conformally invariant expression after integrating over omega
now that would be fantastic if two because a conformal theories are known basically to be
renormalizable and not only that you can compute lots of properties of such theories and they are
very interesting if gravity will be renormalizable that would be fantastic
unfortunately it's not true because yes there are anomalies and anomalies make the whole procedure
breaks down so to see so there's something fundamentally wrong with procedure and now here
my calculations really become nasty probably for some of you but but after we went to start
listening again then so if you work out what it actually looks like in terms of the omega fields
you find that the answer with action i is then multiplied by an omega squared it so happens to
be and actually that was the reason why I wanted this square here because now I get the omega
square in this expression that's very useful because if you're acting this way you also get
a space-time dependent part d mu omega d mu omega so the gradient of omega also then appears in
this action with a very peculiar coefficient six in form that just comes from the calculation
and the matter fields are dependent g hat and omega and the matter fields might or might not
contain a scalar field such as the Higgs particle the Higgs particle could be typically described
by kinetic part of this form but the Higgs field or any other fields in general also carries mass
terms now the mass term is not conforming invariant and therefore mass term contains
explicitly by omega squared terms and now a little trick this is just only a for convenience
I rescale definition of omega by this coefficient in front if you rescale over here you find that
the remaining effect of Lagrangian that you got this way is a kinetic term for the omega field
a kinetic term for the phi field and then things that happen when the phi has a mass
this expression for the scalar fields of matter is a little bit unfamiliar to have matter also
react upon the scalar curvature of space and time but people who are familiar with conformal
theories do recognize the term this makes the phi field explicitly conformal invariant if you
add such a term then you can follow what happens under conformal transformation the term otherwise
is considered rather innocent in flat space time it vanishes if you have no gravitational
no strong tidal forces then this term isn't even observable so it doesn't do very much physically
but now we see that we want this term to make the theory for phi field conformal invariant
the the the part that depends on omega automatically gets this term it came from this action
but now you see something else the omega field which originally played a role as defining the
length scales in space and time has practically the same expression in action as a scalar field
that describes spinless particles in fact omega in a sense describes also a spinless particle
but there are some peculiar differences one is the overall sign this goes with the plus
and that goes to the minus that sign is a long history and it shows there's something basically
wrong with gravity because this all scalar fields must have a minus sign in the action
because it makes the energy of a scalar field positive now this tells you that the energy in
the omega field is negative gravity is an unstable force we could say well we knew it all along
gravity gives you implosion it makes black holes and things yes that's a bit unstable but then
black holes are physically acceptable objects so maybe not that unstable anyway this sign is there
and shouldn't be looked upon too easily it's a very important sign this term here is very
interesting it tells you that the mass term of the phi field has been multiplied by omega
omega tilde and so you get this coefficient in front of that rootless constant sits there
because this transformation the mass squared or the field comes in here but this is now an
interaction between the omega field and the phi field and you remember all together there
are four fields interacting two omega's and two phi's this is a four field action like a vertex
in the final diagrams with four problems coming in and coming out there's no to be conforming
variant of course i've rewritten things saturated theory is conforming variant so not very surprising
that i only get aquatic term here the coefficient in front is not just the mass squared of the phi
field but the mass squared multiplied by the nucleus constant since i use units where h bar
and c are already equal to one only nucleus constant isn't the nucleus constant is dimension
one over a mass squared so if you multiply that by the mass squared of the scalar field this thing
is a dimensionless constant so now and this is how it should be i have now a conform interaction
is dimensionless constant in front so that makes that the effect of theory then comes out to be a
conformally advanced theory that is what i said before and indicating of only itself that doesn't
change that very much however i've not yet considered the anomalies and so so yes this is
this is just a incorporation of the transformation if i now make this transformation and g hat now
become just an arbitrary field i might decide to multiply to drop the condition that the determinant
of g hat is minus one and multiply it by a lot of squared i multiply omega by a lot of minus one
and phi by a lot of minus one this is the way how all these fields transform then the whole lot i
wish i wrote down is completely invariant but now i might decide to first indicate over the omega
field and then something very strange happens um oh you can fix the gauge this is for a black hole
this is for a white hole transform between the two gives you the black hole s matrix a trivial
gauge could be either to have in German g hat minus one or to have omega to be equal to plus one
so there are different ways of fixing the gauge you have to choose one of these
to fix the gauge of theory but otherwise theory has had this complete fundamental invariance
um i could add a cosmological constant the cosmological constant is very much
non-scale invariant if you scale transform the cosmological constant becomes more important
the constant is extremely weak but if you you're on the field is very very large like in the
universe you go to the entire universe or the omega field becomes very big and then this term
becomes important so for the for the universe this term is important for everything else it isn't
so this term sometimes leaves away if i cross it out then actually the integral of omega can be done
exactly and i like to do the idea of being able to do the integral of of something exactly the
reason is that it only occurs protatically in a tangent which means the omega itself obeys
linear equations and when you have such a situation then you can just solve the theory
so we can solve the omega part to integrate it out let's just see what happens
well in particle theory we are quite used to doing such integrals so we know what comes out
if you integrate over the omega field it's just a partial integral not very spectacular already
sort and we all use the fact that the infinite that the integral is actually infinite the theory
doesn't converge there is an infinity and in part of the theory we are so used to renormalizing
everything that everybody else immediately say well it doesn't infinity you just renormalize the
theory we're going to be used to that that's the way it works in particle physics anyway
so let's do the thing in in slight in number of dimensions slightly away from four and then we
can do the integral and then let n go to four you encounter an infinity you're subtracted and you're
dumb so that would be the way in which particle physics will do this but this time there's a
difficulty and that is that when you try to renormalize the infinity you have to go away
from four dimensions but away from four dimensions this theory is not conformally invariant
so renormalizing theory this way breaks conformal invariance and as i said normally usually cares
so we leave that breaking as an interesting phenomenon that you can you can wonder about
but it doesn't disturb you well it disturbs me i don't want that that anomaly so now it's there
and you can compute its coefficients actually life turns out not to be so easy there are more
sources of anomalies the one i just computed actually is normally it only occurs if you have
a curved background field but there's also normally that that occurs when the background
field is flat and i call that the scaling anomaly so if you choose oh my god the same
everywhere you scale things up or scale things down but that transformation also has an anomaly
that is in fact the reason why theories such as qcd uh the uh color force is not conformally
invariant qcd has a completely conformally invariant Lagrangian so naive you would think
the theory doesn't change if you scale well it changes enormously if you scale because
your normally happens to be very big it's a storm attack in theory the anomaly is so big
that when you try to pull the cracks out from each other you get a a very strong force which
scales quite anomalously and that's because there is big scaling anomaly in qcd the same
scaling anomaly occurs in this theory so even when i don't bother about gravity i just go to
a flat background space time i still have a scaling anomaly and i have the conformal anomaly
when the background is unique to zero now now comes the anticlimax of my talk well one of the
anticlimaxes and that is that i can't handle the conformal anomaly when there is a explicit curved
background field and that is simply too hard also the moments are not found to cancel out against
each other it looks at first it looks rather rather straightforward there's only one coefficient
that you have to compute because that coefficient always has the same sign and i couldn't find
anything that cancels it so um so that anomaly doesn't go away but there's another anomaly the
scaling anomaly which i can make to finish and that's a very interesting exercise i go to the
standard model you've seen other pictures of it before and uh consider now this behavior of this
thing under scaling and um now however i add this omega field so i just take g hat to be a flat
background metric that's the concession i have to make because otherwise i can't do it i explained
yesterday that weimberg had made had written down his model for leptons and he would be very very
wise to say it's a model of leptons and not of hadrons because he didn't understand the hadrons
i'm going to follow his his wise example saying okay i can't handle the case and
background spacetime is curved let's keep it flat and let's first try to solve that situation
all i have to do is ask how this theory scales but i do put the omega field in now this is where
the model differs from the standard model the ordinary standard model has no omega field
it just has a young rose field which you can write down this way it has a scalar field such
as the higgs which sits somewhere here the higgs master which may not have a minus sign here
depending on whether the higgs mechanism takes place or not has now a master but it goes omega
squared phi squared the master is multiplied with neutrons constant i write here a k which is a
particular function of neutrons constant k squared is four third part of the neutrons constant i have
a strange one or six in here that's the six which i've been talking about before just to get here
nice coefficient one half this is replaces now the mass term for the higgs it's a conformally
invariant interaction of the scale invariant if you have a far too third interaction that would be
two scales in one scalar arc it's possible in principle that would go linear in omega again
that would be aquatic interaction and the complicated dimensionless by a single power
of kappa fermions it's been one half i written i in the sky by a fermionic part in the engine
those physicists who are familiar with that recognize there's a fermionic Lagrangian
if you have explicit master for the fermions you have to multiply that by kappa such at kappa
times the fermionic the rack mass of fermion again becomes dimensionless it is also closely
linear with omega actually this master for the fermions are just like a scalar interaction
for the fermions normally you introduce mass when not having this this omega field in here
this my explicit mass term but we say that most fermions over their mass due to the higgs
interaction and that means the higgs field phi takes care of it with the coefficient y
that is dimensionless from the start so most of this is the standard model except now i put my
omega field in everywhere and now i say this theory should behave trivially under scaling
because if i now scale g hat then with the omega field i can make it scale backwards so again the
original g mu field will not change at all but the omega field changes and the phi field changes
all this field changes if i make a scale transformation so in principle theory
we could become something totally different and now we're so used to saying that this
when we have a theory of this sort then um uh okay yes here i was i forgot that i highlighted
the things i could be talking about um and uh i claim all parameters with this kappa in the
i dimensionless but this theory is now somewhat different from the standard model because i've
added the omega field in the other difference is that this is supposed to be all the interactions
the i between particles and gravity but you've seen yesterday and in earlier lectures the highway
to the truth in particle physics and you see that you have still many many also managed to go before
reaching the the the Planck scale where all this is relevant and all at all these spots there could
be new additions to the standard model that we don't know about so actually this is not the
ordinary standard model this is the model that replaces the standard model by the time lhc has
reached the Planck scale which will take quite a while so only then that model should be put in
here and also in that model we should put the omega fields and then you should get that this
theory should no longer behave like we are so familiar with the standard model it should
become trivial under scaling again and this is the last really mathematical slide that in physics
we express them by writing down your renormalization group renormalization was the culprit of
everything renormalization was the reason why these anomalies occur but um the renormalization
group tells you what replaces those anomalies if you make a scale transformation which you write
down as a mu d d mu you're saying that mu is a scale parameter and we scale transform all the
coupling constants theory then in principle all the coupling constants all the effective couplings
are scale dependent and it's given by this equation and beta here is a great big function
of all the coupling constants in fact you have as many equations as coupling parameters in the theory
they're not really constant as you as you might realize they actually vary with scale but you
usually call them coupling constants so better to say coupling parameters so let's call them
coupling parameters the cosmological constant the young will's coupling constants the scalar
self-couplings yukawa couplings are two kinds of them a a scalar and absolute scalar yukawa coupling
there could be five to the third couplings there could be mass terms of scalar particles and mass
terms of the spinor particles and all these masses all the couplings they're all now dimensionless
because these are multiplied with kappa and so all these dimensionless quantities change if you
make a scale transformation in theory and that change should be controlled by this beta function
and now we say this beta function is the well-known anomaly it's the anomaly i want to get rid of so i
have to require i have to demand that this beta function itself vanishes and that's something
totally new that this was not done before and it is enormously restrictive because the all these
couple constants i prescribe the scale according to a certain wall which is given by this this
beta is a great big vector it has as many components as there are parameters in the theory
so it means i have as many equations as unknowns if this beta depends on all these coefficients
here all the coupling constants it means that i have to choose every one of them everything
one in such a way that all the beta functions cancel there's n equations with n unknowns the
unknowns are the coupling parameters of theory the equations are the equations that the beta functions
should should vanish and then and i can now tell you these equations are very complicated
equations you can work them out it's algebra it's just algebra there in general this is a
quatic polynomial in the simplest case but actually the higher order connections which are
much worse than that so in principle the equations are straightforward in practice they are quite
difficult they may or may not have isolated solutions right i've n equations and unknowns
well in the complex plane the general any quatic polynomial has roots
but what about the real plane i won't couple constants to be real if they are complex
then the theory would not be unitary and i will have big difficulties if they are real then i'm
in business so but in general the solutions are isolated solutions and that makes the theory very
interesting it means that none of these parameters are allowed to take just any value they are all
constrained by these equations and that makes life very interesting for a theoretician so
we now ask how can we solve these equations there's a whole landscape of different models
um so now you can ask this slide here and explain what have i done because you might ask why was
this not done before and what is the difference between this and all of my theory of gravity
well the difference is that i am demanding this omega field to be a fundamental dynamical field
in the system but this dynamical field can become zero in the Higgs theory or of standard model the
Higgs field can certainly pass through zero without anything happening but if the omega field
passes through zero it would encounter a lower reflex singularity if the beta function wouldn't
vanish and that would not be allowed because that would make that there would be a nasty singularity
in the whole description of the theory so we should demand that that similarity disappears and that's
really the new input of the theory if you would allow a lower reflex singularity of omega then
the condition of the beta function vanishes would disappear and i don't care either any
theory you like but demanding of the omega field is a dynamical field that i should indicate over
forces me to say that the beta function should vanish actually omega is a ghost well it's not
really a ghost it is a dynamical field yes but that doesn't that itself doesn't give it a ghost
it is as it is in principle in order to grab the omega would be an observable field so it's not
just a ghost but it's gross in the sense that you can you can gauge fix it to one much like the
transverse components of the Higgs field are ghosts because you can transform them away
but honestly if i use a different gauge it is a very physical thing so um so yes you so yes you
can you can ask you know how significant is this and i think it is very important that
omega has been kept dynamical and the theory works in that way so um so this is what i
are trying to do it's such a postulate maybe permissible after we made the theoretical form
invariant now so this ends the technical part of story now i want to say something about
about what remains because if too this theory would fix us all copy console all of the system
so naturally um i've tried to see if i if i could find interesting solutions for a standard model
or for other models well i the standard model doesn't work and the reason is it has an abelian
force in it a u1 force replacing electromagnetism and that has always a positive beta function we
know that there's no negative term for that beta function so certainly perturbatively it's it's
always positive so there's little hope of making the standard model conform in variant this way
we have to do some something more drastic look at other models well you think of grand
unified theories which you no longer have electromagnetic sector they have the u1 parts
have disappeared so maybe you have to go to a grand unified version of the standard model
and that would certainly be interesting but it's hopelessly complicated technically and these
equations which you just heard before the uh the beta function equations these equations are so
complicated that um and that i had to simplify things considerably to find whether there are
solutions or not and then working very hard was probably find that you can't find any solutions
it's very very hard it's either quadratic equations whenever i thought i was close to a solution
something else in some other sector went wrong so the problem is to find zeros of these equations
on the real axis of this n dimensional space one quickly finds you need vector fields also
non-immediate type you need uh uh spinner fields to uh for all sorts of reasons any scalars without
spinners without scalars without vectors any but if you need any of those out it doesn't it you
surely won't get zero so we need all these are all these particles and that makes it very complicated
you need quite a few vectors and scalars because the beta function for g will be very large unless
you have quite a number of vectors and scalars floating around that makes your problem quite
difficult but not impossible i said that i started very optimistically trying to solve these equations
find a solution where all couple constants are very small but that turns out to be obstacles
maybe that's a good thing if there were too many solutions you never know which one to choose now i
haven't found any yet so maybe there are just very few solutions at all and those would be very
interesting models of nature but there's another problem and the other problem is now we're getting
very very serious suppose anything of this were true then in the real world we have something
very strange in the real world you have what we call a hierarchy problem and the hierarchy
problem can be can be phrased like this then an astronomer once had a little sun sitting on his lap
and he was experimenting with some the cosmos that he was studying the stars and the planets
and explaining how large planets are and that stars are even larger and how long the distances
are between stars and planets and so on and furthermore the galaxies containing billions of
of planets and such and then little boy looks up as his father and asks him this question
why is universe so big and the father has to admit that he doesn't know the answer that he
doesn't know why is universe so big imagine we had a theory of everything because i'm getting
close to it now look look at the standard model which has always couple of constants fixed
that's pretty much a theory everything but in this theory of everything you just have to wait
and see whether if you press the buttons of theory whether it grows the universe as big as
ours in fact odds are that it doesn't because all to all likelihood the zeros of the beta functions
out of computers way will all be of order one so all the parameters will be order one that means
all masses will be of order the Planck scale or a couple of constants will be of order one
there is not much structure in this universe the universe will be very silly place it will be as
small as a Planck scale and it will then stop doing anything non-trivial beyond that so that's
not the universe we live in by far so there's still tremendous amount of problems the size
universe in meters is very large so now i'm continuing in non-technical terms the last couple
of minutes of my talk and so the size of the universe is very big size stars and planets also
big numbers very different from this size of units in the order of a meter size of atoms
10 to the minus 10 meters in a Planck scale 10 to the minus 35 so compare these numbers 10 to the
plus 36 and 10 to the minus 35 these are you know so big numbers describing our universe
and this are called the hierarchy problem uh so because you can ask you can even argue
that um the dimensionless constants in the universe are the reason why the universe is
so big so you can bring the constants of nature in there to find such a constant is a fairly small
number proton mass electric uh proton electron mass ratio is a fairly large number but the proton
mass compared to the Planck mass is again a universal number in nature and of order 10 to
the minus 20 the cosmotor constant is most outrageous of all this this thing that has
recently been measured at which the Nobel prize is going to be given that thing is 10 to the minus
122 so why is this constant 10 to the minus 122 compared to that constant what is it it's a
dimensionless number that nature just gave us for no particular group reason as far as we are concerned
so but it is very important otherwise if these numbers were only like that the universe could
not possibly have been so big for instance we will try to to calculate size of things in terms
of fundamental constants now this is a lot of of hand waving so i'm not very accurate let's say
that the size of nucleus is roughly h by all the the quantum wavelengths is a proton it's not quite
right it's the quantum wavelength of the pion the pion is only one tenth of the proton mass
that's why i put a factor 10 in front but um it's the orders of magnitude that count here
size of atoms contains the combination of the electron but then also to find such a constant
in there because the electrons held together to nucleus by the find such a constant by its electric
charge which is a number of 137 so that's why atoms are so much bigger than the nuclei it is
a consequence of the fact that this constants vary so much now if with the density of rock
i can express that in terms of of natural numbers more or less the size of atoms to the third power
and then related to the weight of the proton you get 11 grams of cubic centimeter that's in order
of magnitude not so bad if you look at how large these numbers are and the mass of the planet
where i define a planet or something that can hold its atmosphere and the atmosphere is basically
held together by chemical there's chemistry going on in the in the atmosphere but not so
much that the atmosphere evaporates so when you calculate how much that masses be well you got a
little more than the mass of the earth so somehow the the temperature of the atmosphere is a little
below the temperature where most chemical reactions take place that's why the earth can still have an
atmosphere so otherwise we'd have to be a bit heavier like like jupiter or so anyway jupiter
is a typical planet of that sort then you get the mass of a star well star let's assume that
the atmosphere of the star has nuclear reactions going on then you get a factor of 400 000 bigger
than the mass of a planet so you can understand well the size of universe imagine what is controlled
by the by the cosmic constant actually this is not quite right here i'm certainly using a lot of
hand waving here but you can roughly understand why the universe contains large numbers if you
can understand why the constants in nature have such enormously variable values that you at least
you're halfway solving the hierarchy problem but only halfway because now the question is why do
these constants generate such big numbers or such small numbers and this i call the real hierarchy
problem one person who tried to answer this question was Paul Dirac Dirac who became famous
for his theory for quantum mechanics and the electron so on later in his life
you know i only met him a few times was later in his life he was advocating this theory his
answer is very simple the universe is so big because it's so old and that was his answer
he said and then so how could it be well suppose that all problems in nature are time dependent
so imagine universe is is 10 to the 61 fundamental time when it's old that's very old but suppose it
comes of nature depending on time now you can put here a time dependence of this sort you can
imagine one of our alpha to be logarithmic depending on time the polterian plant mass
ratio could be this function then you get the right orders of magnitude so in the only thing
you have to explain them is where these expressions come from and of course you don't know
but you can guess them and say well if this is the answer i can understand how this hierarchy
problem solves itself in a way unfortunately the theory could be tested or unfortunately
theory could be tested because now but it was found to be wrong you can make action nowadays
there are many ways of measuring these things for instance alpha the time dependence of alpha
is much smaller than nira could predict if you calculate how a log of the age of the universe
depends on time you'd find a much bigger alpha time time dependence of alpha but this over 10 to
the minus 14 per year this numbers like this were were found by looking at the with the frequency
comb this very beautiful way of measuring very accurately certain constants of nature
and we can do even better or this is the frequency comb with the frequency comb
you can get 10 to the minus 17 per year so those numbers are colossal and they really contradict
in a big way also this is an interesting experiment the astronauts polar astronauts left
a device on the moon and this device is just a reflector it's a back reflector it reflects light
back from where it came from so if you shine a laser light on it the laser beam is reflected
back in direction from where it came and this way you can do lunar ranging you can very accurately
measure the distance between the earth and the moon by with this instrument and it is various
instruments of the source so you can measure very precisely how far the moon is from the earth
and more precisely you can measure whether this distance between the earth and the moon changes
with time well it does change in time but you can calculate how it changes in time by doing the
planetary dynamics of how plants in and the sun and the earth or influence the earth moon distance
but this can only be calculated and checked with the theory if you find a deviation that's
interesting what would the deviation be caused by well it's reason to assume that if the
gravitational constant changes with time then a total angular momentum of the system should remain
constant and you're going to ask very carefully what does this mean well people say they measure
the time dependence of neutrons constant but that's not what this measures it measures the
time dependence of a scale invariant of a dimensionless quantity actually i claim this is what you
measure the time dependence of g times total mass of the third divided by h by times the electron
mass and the reason why is these combinations is the the total mass of course determines the
masses in the problem with the electron mass determines how fast our human clocks are actually
running because most class use electromagnetic phenomena to to measure say spectral lines of
atoms but that always contains the electron mass so basically this is an dimensionless quantity
that you end up with measuring and that number is naturally 10 to the minus 35 so according to
Dirac that should be time dependent but actually this changes less than 10 to the 12 parts
one part 10 to the 12th per year and Dirac will give a much bigger prediction so unfortunately
Dirac's theory is wrong but i sympathize with this beautiful attempt to find an explanation
so what is the explanation of why number has to be bigger so small and i yes i mentioned
yesterday what the explanation could be the so-called landscape and well somewhere in this
landscape is our universe and we don't know which and maybe nature just chose the numbers to be very
large because otherwise they hope we know living beings to marvel at our universe and okay you can
take that as an argument or not and we don't know how serious it is but it could be the answer
that simply there are only very few spots on the landscape where where the zeros of the beta
function are very very special very very large and that's or small and those numbers are numbers
that we are looking for i sincerely hope that another more interesting explanation can be found
but i'm very worried that this argument might be correct so according to some strengthiest
there are 10 to the 500 different universes in the landscape and if this number 500 is bigger
than the 122 of the set of the cost of water content then there will be many solutions that
actually allow for these very wildly varying constants of nature and so the verbal universes
in which people can can walk around and and study our universe but this is what i wanted to
the message i want to bring across how possibly studying black holes can have an effect on
on how to look at elementary particle interactions so that's my talk of today and this ends also my
uh my uh my for my second lectures thank you
thank you very much
well the sector which i studied is renormalizable so if your background space time is flat
then the remainder the omega integral happens to be renormalizable so immunization counters are
only becoming significant if you take curvature of the background so that is why i said i could
only study part of the theory without the whole theory because of course gravity is not renormalizable
so the curve background i can't handle but the scalar part happens to be something you can
discuss so this is why i think this is not the ultimate theory the ultimate theory should contain
everything including curved space time of course well that i can't do but in the meantime i could
check whether this approach gives me a partial answer to the questions i have
and it remains to be seen whether it works but at least it's it i think it's an interesting attempt
it depends you can as i said it's a gauge theory in the same sense of whether the Higgs
field is a propagating field or not the length of x field yes but but the the launch the the
transverse parts of the Higgs field in sense also non-propagating or propagating depending
on how you fix the gauge well um one can argue about this uh well of course you know that if
you add if you take the the g hat field as well the fundamental degrees of freedom you get are
the gravitons so gravity has two fundamental propagating modes which are left to the right
hand at the graviton right or the two polarizations of the gravitational field so so that i think we
agree about and oh my god doesn't give anything else than that but um if i fix g hat to be zero
i only fix nine of the ten components of the metric tensor to be zero the tenth is something
that still now propagates so um so i put a rather strange perhaps constraint on the theory
that allows only one of the gravitational components still to propagate it's not really
the graviton it's some funny funny combination of the gravitons um but um
well yeah um oh i had i think i have an argument in here if i can find it
there's something that was set up no this is not it
let me try to
see if i can find
um no no i have to
search what i had a slide here this is slides i haven't shown but um
i thought i had a slide here which is relevant for that
um
no i i can't find that slide back there was an argument oh yes here it is um that if you um
um consider the um omega field contribution with the wrong sign you can actually observe
that after having made the split but the omega and g hat g hat no longer has a kinetic term
and it's Lagrangian so um the Lagrangian then would contain an l depending on g hat an l omega
but the l of g hat vanishes so you only get an l omega and l matter so now if you vary this with
respect to the g hat metric then you find only contribution from here and a contribution from here
this gives the the meta field gives the usual einstein tensor divided by magnitude constant
the omega field gives a similar contribution from omega and now varying these two together should
give zero that means that the omega generates an energy momentum which is the wrong sign and
which cancels out any momentum of matter so the omega field is not truly unstable but it matches
the energy momentum of all the matter fields so you find and that is of course einstein's equation
so einstein's equation tells you that the matter energy and mental tensor cancels out against the
energy mental tensor as generated by the einstein tensor but this einstein tensor actually is
represents the energy momentum carried by the omega field so it has the wrong sign just in order
to cancel that but since the total has to be zero there's no instability because of that
so whatever the matter energy mental tensor is that's cancelled by the gravitational point
and that gravitational point is due to the omega field because this thing there's no contribution
at all not in the tree at the tree level say quantum mechanics is more is a more complicated
story but at the tree level g hat doesn't contribute any to the total energy mental tensor
and equation is just this einstein equation where the einstein term represents any momentum carried
by the omega field but having the zero here means that the omega field cannot just generate any
energy and mental tensor likes it must cancel out this part
we have a standard model that is free to function i think but the zero is not satisfied
of course yes so can you use this as a diagnostic tool that tell us what kind of
uh
i wish i could answer that question because i did ask that question myself can what what can i
do starting from the standard model and i'll say well now we have to add more terms to make its
beta function cancel is there a systematic way of doing this and i have not been able to find a
systematic way of doing this so that's why i can't answer the question but i i wish i could yes
so
i should have mentioned that i forgot to mention that that there is an example there is an existence
proof well not quite i can take no you mentioned something good i forgot to mention that there is
a solution and that is called super young mills and with n equals four super young mills is a very
super symmetric theory and if i now say there are no masses in the theory then the omega field just
drops out so and i take it in the flat background space time that theory is conformally invariant
all by itself so if i take g hat zero then i can just add this n equals four super young mills
theory and then indeed that is a solution but it's a very degenerate solution and very special
solution and to my mind also very unphysical because it has no masses at all and it has a
complete conformal invariance that's too much so so although it is a form of the solution it's not
the solution of the kind that i would find interesting but it could be the starting point
of a more general construction as just uh the other gentleman asked so maybe that maybe there
is a way to to arrive at the more interesting by starting from n equals four super young mills
if you add very small terms in the side see if you can do something well and then if there are
no masses that may be not a problem because they could be generated by spontaneous symmetry
well actually it could be an advantage having no masses because the masses which you do want to have
are very very small at the Planck scale like 10 to the minus 20 or so so think of n equals four
super young mills being a very good but not perfect approximation of the solution it has
deviations of the error of 10 to the minus 20 that would be great because that's what i'm looking for
so so we could start at n equals four super young mills and then add one field or something
uh that breaks the conformal invariant explicitly but then in a in a controllable way
and that last sentence is very very important because usually it's it happens in an uncontrollable
way and then you don't know what to do anymore but suppose you can say well we are nearly there
but now let's make a very small change
i have to say perhaps the question is a little bit stupid but i would like to ask
at the beginning of the talk you compare the two observers one is inside the hole and one's outside
and okay when when the outside who's sending what's a release looking to the other one falling
in then due to the red a redshift he stops at the surface at the horizon for the duration of the
universe yes so if the universe exists the other observer is on the surface so if i twist this
implication on it means if he's not on the surface if he's inside the universe does not exist anymore
or the black hole does not exist anymore because a black hole radiates way after some time
no it's it's not a stupid question you have to ask questions of the soil all the time
but the question was can there be a a more rigorous remedy to the problem what i really
wanted i couldn't don't care so much about observers what i do care about is a consistent
physical theory and so what i mentioned later in the talk was this called so-called scattering
matrix the s matrix saying that suppose you know everything that goes into the hole no matter what
happens things will come out so suppose you know everything that goes in you should be able to
compute everything that comes out that is the fundamental requirement and that i just translate
in words as a government entity principle
so ladies and gentlemen we've reached the end of this year's unfettered von Reizek
lectures i would like to thank professor toft for giving these very impressive and for many
of us very challenging lectures um i think it was a good idea to have a form for these lectures
that provides both for three introductory lectures so to speak and one lecture that
addresses persons with a much larger background in physics so that both physicists
about physicists then learn a little insights from these talks and we are very grateful to you for
getting these lectures here thank you very much