Quantum Field Theory and Gravitation
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00:00
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Transkript: Englisch(automatisch erzeugt)
00:16
Yeah, thank you very much for the introduction and also for I would like to thank all the organizers
00:22
for inviting me to this nice event. I want to speak about the relation between quantum field theory and gravitation. So this is based on joint work with Romeo Bronetti, Michael Deutsch and Katarina Reisner. Now we all know that the problem of relating quantum theory and gravitation
00:54
is a long-standing issue around 100 years old and without a definite solution until now.
01:01
In spite of the fact that quantum physics is a well-established theory with a lot of nice verifications, in particular also in elementary particle physics where of course the mathematical status is not as good as in non-relativistic quantum mechanics.
01:34
But nevertheless it's applied with great success to experiments in elementary particle physics.
01:44
And on the other side also gravitation is nowadays well established as a classical theory. There's a lot of experimental confirmations but we have not yet a consistent theory which combines both theories. Now here in my talk I want to approach this problem from a rather conservative
02:07
point of view, namely from the point of view of quantum field theory. And so the first step is then to look at quantum field theory under the influence of external gravitational field,
02:22
which mathematically means that you have to study quantum field theory on generic Lorentzian manifolds. And I will then discuss in which sense this can be also used for including perturbative quantum gravity. So when you start to define quantum field theory
02:50
on curved backgrounds, in the first moment one might think that this is a very simple problem because in classical field theory everything is defined in terms in geometric terms and so it's
03:04
rather easy to formulate classical field theory on generic manifolds. For instance in ectodynamics you just need the notion of differential forms, exterior differential, and you need the Hodge tool to define the co-differential.
03:35
So you see you need only very very few things, just that you have a differential
03:41
manifold and that you have a metric which is then used to define the Hodge tool. Now when you try to do something similar with quantum field theory you see that the standard formulation of quantum field theory uses a lot of other structures.
04:00
Namely you define the concept of particles in terms of irreducible representations of the Poincare group. You assume that there exists a distinguished state, the vacuum, which is defined as a state where all particles are absent. We have as a main physical observable
04:20
the S matrix which describes the transition from incoming to outgoing particle configurations and we use as a technical mean the momentum representation which is possible due to the translation invariance. And as another tool we use the transition to
04:43
imaginary time so that we get a theory on Euclidean space. And when one tries to put quantum field theory on a curved background you see that none of these features is present.
05:01
So this in the generic case the group of space-time symmetries is trivial so there is no useful concept of particles on a curved space-time. Therefore there is also not a notion of a state without particles so the vacuum is no longer a
05:21
valid concept. The transition to imaginary time so that you can replace your Lorentzian manifold by a Riemannian manifold is not possible in general. You cannot do calculations on momentum space and if you just close your eyes and you write down Feynman graphs then
05:44
you meet the problem that there's no unique definition of a Feynman propagator. Oh and actually these theoretical obstacles are then seen in some some effects. What was
06:04
first observed was that there is a phenomenon of particle creation so in three theories you might find the definition what you call a particle but then you see that this depends on the choice of a Cauchy surface. So when you change your Cauchy surface and you get a different particle number so you have particle creation from Cauchy surface to Cauchy surface
06:26
and actually even infinite amounts of particle creation so this shows that the particle concept is not meaningful. And there's this phenomenon of Hawking radiation so that if you look at a collapsing star and you start from a state which is sufficiently regular then you see that
06:45
asymptotically the state develops thermal radiation. And even on Minkowski space you have this problem if you look at an accelerated observer and then you see some thermal properties of the state the so-called Andrew effect. So in order to define quantum fields in a curved
07:10
space-time it's important to decouple those features which are geometrical from features which are non-geometrical. So what are the geometrical features these are field equations
07:24
these are commutation relations so in general these are the algebraic properties of the theory and this forces one to use the formalism of algebraic quantum field theory as developed by
07:41
Araki and Kastner. Of course these so you have your physical quantities as elements of an abstract algebra of course this algebra should admit representations on Hilbert space but you don't use a specific representation. There is however one ingredient of Hilbert space
08:05
representation which is very important for the structure in quantum field series this is spectrum conditions the conditions that the energy is positive. And so what you need in order to do this program is you have to find some local version of the spectrum condition. Okay so this
08:26
is the plan of the lecture so I will give few remarks on Lorentzian geometry and field equations then I will discuss quantization in the sense of deformation quantization. The local version of the spectrum condition is imposed by using some tools for micro-local
08:49
analysis and then I will discuss how renormalization can be done in this framework what the notion appropriate notion of covariance is and I discuss this on the example of a scalar
09:05
field but then I will indicate how this has to be generalized to include gauge theories and even gravity. Okay so let me start with some basic notions so we have a smooth manifold which we call a space-time and space-times in my talk are always Lorentzian space-time so
09:25
the metric has Lorentzian signature. We have causal curves which are tension vectors which are always time-like or light-like. We have time orientation which is imposed by choosing some nowhere-vanishing time-like vector field and so we can say when
09:47
the curve is future directed. And then we define the future of some point x as a set of points y
10:01
which can be reached from x by a future directed causal curve and in the analogous way we can also define the path of a point. Now what is important for the following structure is the notion of global hyperbolicity. So space-time is called globally hyperbolic if it does not
10:26
contain close causal curves and if for any two points x and y the intersection of the past of y and the future of x is a compact set. And the nice feature of these globally hyperbolic
10:44
space-times is that they have Cauchy surfaces so hyper surfaces which are hit exactly once by each non-extendable causal curve and they even have a foliation by Cauchy surfaces so they have a rather simple structure so they are diffeomorphic to a product of
11:06
some manifold sigma times the real axis. Particularly globally hyperbolic manifolds are never compact. And the most important feature is that normally hyperbolic linear partial
11:27
differential equations have a well-posed Cauchy problem. In particular they have unique retarded and advanced Green's functions. Actually these properties of globally hyperbolic
11:42
space-times are around for quite some time but the proofs are not so old. The complete proofs are due to Bernal and Sanchez but you find it already in older books the same statements.
12:01
Now let's look at a simple example the Klein-Gordon equation and there we have the retarded and advanced propagators as maps from the space of test functions with compact support two spaces of compact of two smooth functions. So D is a space of test function with compact
12:22
support and E is a space of smooth functions. And they are uniquely determined as the inverses of the Klein-Gordon operator and by the support conditions that the support of delta r
12:40
f is contained in the future of the support of f and vice versa with the advanced propagator. Now what is crucial for the structure is that these two propagators are different. So we have a difference which is often called the causal propagator and later named the
13:03
commutator function. This will be crucial for the algebraic construction. Namely in terms of this commutator function you can define a bilinear form which is anti-symmetric and this bilinear form on the test function space and which is degenerate but
13:25
the degeneration is of a very simple form namely if you these form vanishes if one of the entries is in the image of the Klein-Gordon operator applied to a test function.
13:43
So doesn't it look symmetric the Laplacian or the all the other functions? No no it's anti-symmetric because the adjoint of the of the retarded propagator is the advanced propagator so just if you take the adjoint just the role of advance of retarded changes and because it was a difference it changes the sign.
14:07
Oh it's not the Laplace operator oh I'm sorry it's not the Laplace operator I'm sorry this is a convention yeah it's just a commutator function no no okay now now this fact that
14:22
this is the anti-symmetric can be used to define the Poisson bracket of classical field theory which is just the following that's the commutator the Poisson bracket of fields at points x and y is just this the integral kernel of this operator so it's just this
14:43
distribution delta of x y. So is it integral on x or is it like f of x and then g of x and g of y like like no no delta okay the convention is that here oh I'm sorry delta is already an integral
15:08
delta is an operator which maps the test function to a smooth function and the result is again a function of x yeah so it's but but here I write a delta of x y which is the integral
15:21
kernel of this operator okay now I come to quantization so we have the space of field configurations for scalar field and as this space we use all smooth functions not only solutions all smooth functions and we define observables as functions on the space of
15:48
configurations yeah so so it's off-shelf it's a so-called off-shelf formalism which is of course one could restrict oneself to to a solution space but this is usually
16:02
more complicated and it's for the quantization turns out to be much more convenient to use the offshore formalism yeah and maybe observables with inclination marks okay I don't want to discuss a measurement problem yeah it's just I call it it's just a name at
16:22
the moment yeah but but I think it's actually the observables are constructed in terms of these objects yeah and which how they can be observed is a different question I want don't want to discuss so these are just maps on this on this space of configurations and of course I am rather
16:47
rather huge set so let us look at special cases so for instance in Weidmann field theory you are used to use linear functionals so you just integrate the field phi with some
17:01
test density f and you can then of course multiply these functionals in the sense of point wise multiplication so you get polynomials and a slight generalization is to look at polynomials of the field in this way that you have the field at different points and you
17:23
integrate it with the test density in n variables and an other class of functions which is very important because the interactions are formulated in terms of these functions are the local functionals these are just defined in terms of a function on the
17:46
jet space of m and as a technical tool we need differential differentiability properties of these functionals so the notion of a derivative is very simple namely the nth derivative
18:05
is just defined as a directional derivative and we call the function differentiable if this nth derivative defines a symmetric distributional density and there is some
18:21
further condition on the dependence on the on the field configuration phi which i don't want to discuss but i think this is a nowadays a well-established framework of infinite dimensional analysis psi is in this case a smooth function it's not not necessarily compact support but i
18:41
assume that the the functional derivative is a distribution with compact support okay now i can extend the Poisson bracket to the space of of this larger set of functionals by the following
19:01
formula so the Poisson bracket of two such functions capital f and capital g is just given in terms of this commutator function integrated with the functional derivatives of f and g and this has all properties of a Poisson bracket and was originally be defined by payals so-called
19:22
payals bracket now we want to quantize this structure so we use a concept of deformation quantization which means that we look for an associative product which depends on the
19:45
the limit h bar to zero and the commutator divided by i h bar approaches the Poisson bracket and there's a simple solution for this in this simple in this in this case name is
20:00
the Weyl-Meyal quantization where you define the star product by this formula here so you apply a formal power series of bi-differential operators to the product of functions at different points and then at the end you set phi one equal to phi and to phi two and this is to be understood
20:28
as a formal power series in h bar but of course in case your functionals are polynomial this is a finite sum yes also for the polynomials this is an exact definition but in the general
20:42
case this is only a formal power series so let's let me describe it on a case of a linear functional then the star product is just the classical product plus in first order an h bar
21:01
a term i times half of the commutator now what is bad with this Weyl-Meyal product in quantum field theory the bad feature is that if you want to extend this product from the linear functions to non-linear local functionals then you get problems namely let me discuss it
21:29
in the simplest case of phi squared so phi squared integrated with a test density f is a well-defined functional it's also differentiable everything is fine but when you apply this formula you get
21:43
three terms the terms without derivatives it's just the classical products then you get the term with one derivative which is okay but then you get the terms with the second derivatives and there you have to square this distribution delta but delta is singular and general products
22:04
of distributions are not well defined and actually in the case of delta you cannot help there's no reasonable way of defining this square as a distribution but now the positivity
22:21
of energy helps us as we know from ordinary quantum field theory and we can also do this in this more abstract framework namely we use the fact that this condition on deformation quantization do not fix the star product we can replace the commutator
22:42
function is this exponent by any distribution h whose anti-symmetric part is i times the commutator function because only the anti-symmetric part is tested in the commutator and if we do this then we get a star product which we are called here on the slides the star h product
23:04
which is equivalent to the previous one namely let us look at the operator gamma index h which is defined as the exponential of h bar over two times the distribution capital h
23:28
paired with the second functional derivative and then so this is again defined in sense of formal power series and you can then easily check that the two star products the original
23:43
Weimar Jahl product and the star h product are related by this operator and also this operator is invertible so this is really an equivalence of star products but we would like to find a distribution h such that we can extend the star product to more singular
24:09
functionals so we have a certain wish list for for this function h so we first we require that it's a bi-solution of the Klein-Gordon equation this is just nice because then
24:23
the functionals f which vanish on solutions form an ideal in this product with respect to this product as in the classical theory this is very nice because at the end we would like to go to the Onschel formulism where we divide all the ideal of functions vanishing on solutions then in order to have a well-defined product also for local functionals we
24:47
want that point wise products of these distributions exist furthermore we would like to have something which is definitely wrong for the Weimar Jahl product which which can be required here
25:05
namely we will require that h is a distribution of positive type which means that the integrated h this the test density f and these complex conjugate and test density f bar
25:20
then you get over the non-negative number but h is not symmetric h is not symmetric no eight but actually what enters here is the symmetric part of h because you have the same function
25:41
on both sides yeah so so that's really positive this is a requirement and what is the consequence if you use such an h product then you have the nice feature that you immediately get a lot of states states on this algebra are functionals which are positive in the sense that they assume positive values on products of f bar f and if you have this property then you can easily show
26:09
that every evaluation of this function at a field configuration yields the states these are just the coherent states so this is a very nice feature of this one it's a little bit like when you pass to a k-layer to a k-layer structure it's related to the k-layer structure yeah that's true
26:38
and furthermore i have this requirement on positive energies namely i want this function h
26:45
selects locally the positive frequencies and okay these conditions don't come from the sky it's just what we already know from the two-point function and minkowski space so we just want to save as much as possible from the situation we know minkowski this there's
27:05
a whiteman two-point function exactly fulfills all these conditions and actually this transition to this new star product just amounts to vic ordering it's just civic ordered version yeah could you also select locally equilibrium some kind of local kms condition also then the
27:25
positivity of energy actually the kms states with positive temperature also satisfies this condition so instead of the whiteman two-point function you could also use the two-point function of a kms state they have to fulfill all these conditions so locally there is no
27:42
difference between the positive energy and the energy condition on a kms it is positive temperature
28:00
yeah okay so now i can this to discuss this condition when uses techniques from micro-local analysis in particular when uses the concept of wavefront sets so the wavefront set is a subset of the cotangent space so the the commutator function delta
28:21
is a distribution on the product of the manifold with itself and the wavefront set is then a subset of the cotangent space of the product of the manifold with itself so there are two points of the manifold x and x prime and at each point you have a covector
28:42
k respectively k prime and the wavefront set of delta just because delta is a solution of the plan gordon equation can be shown to be of the following form there exists a null geodesic connecting x and x prime and this covector k is co-parallel to the tangent vector
29:03
of this geodesic at the starting point and then you take a parallel transport of your covector to the other end of the curve and you add it to the other vector this must give zero so this is some kind of translation invariance or momentum conversation conservation
29:22
if you want this is the wavefront set of delta and if you look at the wavefront set of the positive frequency part it's just half of this the positive half of this wavefront set so you just have the additional condition that's the momentum or the covector at the point x
29:40
is future directed and actually it's clear because the sum if you have a sum of two distributions of wavefront set is contained in the union of these two wavefront sets so so it cannot be smaller because when you have this condition that two times the
30:02
imaginary part of delta plus is just a commutator function then this is the smallest possible wavefront set of course you could also think of the negative sign in this relation this would be give the delta minus function but delta minus because of this minus sign is not a positive type
30:22
so what you see is that these two positivity concept positivity in the sense of quantum mechanical probability and positivity in the sense of of the spectrum condition the positivity of energy coincide for the scalar field i know of no fundamental reason why this must be so
30:46
actually this is a great problem in defining three fields on curved space time this half integer spin higher than one half so there you get get a lot of problems because this positivity issue is usually not there seem to be no reason
31:06
that this is related but for the scalar field it is related for the dirac equation it also works fortunately okay now this uh uh concept of the wavefront set can then be used to define
31:28
what is called the hadamard function this is due to ratzy kofsky and uh so uh a hadamard function is a solution by solution of the quan gordon equation
31:42
such that the two times the imaginary part is the commutator function and the wavefront set is just this positive half of the wavefront set of the delta functions is then called the micrologal spectrum condition and h is of positive type but the waveform set is always pointing up in the fracture and not yeah for it's it's for the for the first entry
32:07
for the first entry yeah but it is no space like vector no space like yeah yeah yeah yeah it's uh actually it's light like slight like but future directed actually this concept of the
32:23
hadamard function is older than the work of ratzy kofsky and uh you can just try to find more explicit form for the hadamard function so this form is given here this is the form u divided by sigma plus v times log sigma plus w where u v and w are smooth functions and sigma
32:44
is just the squared geodesic distance between two points and what ratzy kofsky proved is that his definition is equivalent to this definition actually this definition has to be made more precise because you have to to discuss the singularity in light like and time like directions
33:04
you have to say what happens if you are outside of the of a geodesically convex neighborhood and so far so this is a very complicated definition which was fully spelled out in a paper by k and walt and it's a major technical advance that that this can be reduced to this simple property
33:27
of the wave front set so this actually this definition of hadamard solution in terms of micro-local analysis makes uh it uses a lot of things which could immediately be done so
33:48
first this was a construction of big polynomials on curved space them as operator value distributions so because you just compute correlation functions of big product everything can be done in terms of
34:02
this distribution age and its derivatives and you just have to see that all the products which arise are well defined and this can be done in terms of the wave front set the condition is that the sum of the wave front set of the factors never should hit the zero section of
34:20
cotangent bundle and this can be it's just a consequence of this positivity condition on the on the co-vectors for instance in this way you can prove that in the hadamard set not only the expectation value of the energy momentum tensor is well defined but even the correlations
34:41
are well defined so so so if you want to understand the back reaction by looking at the expectation value of the energy momentum tensor you know of course you have to also to look for the fluctuations because otherwise the expectation value would be not of much uh much use other
35:03
aspect are the quantum energy inequalities originally proposed by a fault but fusor found that this micro local spectrum condition gives a very uh important generalization
35:21
namely what one finds is the the expectation value of the energy density in the hadamard set is not necessarily positive this is already true for the energy momentum tensor in minkowski space but when you smear the energy the energy density with a square of a real value test function then it
35:44
becomes bounded from below and this can be shown in terms of these wave front sets and maybe the most important thing is that you can do using these techniques the ultraviolet renormalization of quantum fields during curved space time and this was done by bonetti myself and then completed
36:03
by pollens and walt infrared is a different issue this is a different issue okay so how is renormalization done now uh the method is based on the concept of causal perturbation
36:21
theory as originally proposed by stickelberg and bogalubov then worked out in all details by ebbstein and glaser so the basic idea is that you define the time ordered products of big products of free fields as operator value distributions on fox space and you require that these time
36:41
ordered products satisfy a few axioms and the most important is that the time ordered product coincides with the operator product if the arguments are time ordered so what you would usually require and then in the famous paper of ebbstein glaser from 1973 they succeeded in
37:05
proving that solutions satisfying the axioms exists and that the ambiguity is labeled by the known renormalization conditions and then you can construct the solution either directly which is in many cases somewhat complicated or via some of the known methods bphz
37:23
polyvelar momentum cut of dimensional regularization of what cells now we wanted to apply this to curved space time this required a reformulation which only uses local concepts
37:42
this for this was in a flat space yeah now if we do it on a on a curved space time so we cannot use all the techniques so we have to generalize this a little bit so we have cannot have not this distinguished this representation of the fox space so we do it space free yeah
38:01
without any reference to fox space we don't have translation symmetry and we need a new concept of the adiabatic limit because just makes no sense to take the integral over all space time if you don't know what the situation at the end of space time is
38:21
and you need renormalization conditions which are in a sense universal i will discuss what this means formally what you do is you construct an operator called the time ordering operator which is formally given by this formula so you
38:41
take the Feynman propagator associated to capital h which is just h plus a multiple of the retarded propagator and you have this pairing of this distribution h f is a second derivative and okay i omitted here factor h bar yeah so you should think of a factor h bar and the factor
39:06
one half and now the question is how this is of course rather formal so you have to to see whether this can be made precise and it can be made precise as a map from the set of multi-local
39:25
functionals to the so-called microcausal functionals let me briefly explain what this means so we have the local functionals we restrict the local functions to those which vanish at phi equal zero so the constant functional is not included there and then the multi-local
39:46
functions is just this is just a unit algebra generated by the local functionals and the Michael causal functionals are those functions which where the wave
40:01
satisfies a certain condition of course this explains this condition in detail is maybe a little bit too complicated at the moment just remember that it's a condition just made in such a way that the products which arise are well defined just as an example so we apply this
40:22
operator to a simple multi-local functional which is just a square of two functionals of the form squared integrated with the test function and then we have applied this operator so we get three terms just the first term which is a unit then the first derivative and the second
40:45
derivative and the problem is again in the second derivative because there we have the square of this Feynman propagator hf and but the Feynman propagator is not not a solution of the homogeneous Klein Gordon equation it's it's a propagator so it contains a
41:08
wavefront set of the delta function which is just the diagonal so the the wavefront set of the delta function is just the diagonal x x so two points should coincide
41:21
and the co-vectors have to add to to zero but k can be arbitrary all all k's are admitted all non-vanishing k's can be admitted and then of course there is no positivity condition and so when you define the square you can add two co-vectors and you get zero
41:45
and this is just shows that the in general this is the the square is not a well-defined distribution
42:00
but this holds only at coinciding points at non-coinciding points you are always in the situation then that you can use these positivity condition so this product is well defined for non-coinciding points but not for coinciding points and this holds very general and
42:23
so what remains at the end this is already in the framework of epsilon and glass and the general case can be reduced to this is the following mathematical problem you have a distribution which is defined outside of a sub-manifold and you want to extend it to
42:42
the full manifold and you use something corresponding to translation covariance on curved spacetime which is a technical problem but can be solved and so at the end you get a distribution which is defined everywhere outside of a single point
43:04
and then you can discuss this extension in terms of the scaling degree which was originally defined by steinman so this is the following definition you just scale your distribution
43:21
by some positive factor lambda you multiply the distribution by a certain power of lambda and you ask that this sequence of distribution converges to zero in the sense of distributions as lambda goes to zero and as lambda goes to zero yeah and so you test the the singularity
43:47
as the origin and and the infimum of these numbers is called the scaling degree and then the theorem is that if the scaling degree is finite then extensions exist and moreover
44:02
extension with the same scaling degree and two such extensions differ by a derivative of the delta function of order scaling degree minus n so in particular the scaling degree is smaller than n then the distribution is unique and if the scaling degree is infinite then
44:23
a t cannot be extended so this theorem i think this is due to to way yeah okay i'm not sure who first gave a complete proof of this so we were not able to find a complete proof so it would be nice to know it i think okay
44:44
okay also for many sub manifolds ah yeah okay okay so so that's uh so this theory replaces completely the standard regularization techniques
45:01
but in order to get a specific extension they are often useful and uh just to explain how this is related to standard regularization techniques um the following considerations so
45:21
let's assume we have a finite scaling degree but larger equal to the of the dimension and then one can show that the distribution can be uniquely extended to all test functions which vanish at the origin of the order of the degree of divergence which is just the
45:41
integers largest integer smaller than the difference of the scaling degree and the dimension then we choose a projection on the complementary subspace of the test function space and then every extension is given by this formula so we just take the composition of
46:03
the distribution with the with the projection one minus w w is this projection the finite dimensional subspace and one is the unit operator and all extensions are of this form now such a projection has a simple form that it's can be written in this
46:25
direct bracket notation as a sum over w alpha delta alpha del alpha delta so the functions w alpha form a basis of this finite dimensional subspace which is just dual to the to the basis of the distributions
46:47
arising from derivatives of the delta function so this means just that the derivative of w at zero is the Kronecker delta times some sine now assume we have some regularization techniques so
47:03
we replace t by some sequence tn and we assume that tn converges to t on the subspace of test functions which vanish at the origin then tw was defined as this composition of t this
47:21
one minus w now i replace t by tn and then i can apply tn to both terms so first i apply to to the unit and then i apply to this projection w and what we see is that we can construct this distribution as a limit of tn where we subtract certain divergent
47:47
multiples of the delta function or its derivatives this shows how divergent counter terms occur in this formulism so it's just because you have this projection but you cannot split it for t itself but only for the approximating sequence tn so i guess yeah there
48:03
are two ends on your slide is different meanings oh i'm sorry yeah here this is the okay so so these ends are not related yeah but your space time is now an island no one
48:25
no no so this was just the general comment of course this is some technical problem to reduce this problem of extending distributions on manifolds which vanish on a which are not defined on a submanifold to the situation where you have just one point so in the sense
48:44
you use some transversal coordinates near to the submanifold yeah that's the way it can be done but this is technically technically demanding but i i can so this was done in this paper by
49:00
bonetti myself and there's a recent phd thesis by viet dang who analyzed this from the mathematical point of view very carefully but at the end you arrive at this problem but this of course requires some exercise in micro-local analysis okay now we have all the means to
49:23
construct this time ordering operator so we look at the symmetric fox space over the space of local functionals so fox space no just in the algebraic sense so it's not considered to be a space just the sum of symmetric tensor powers and on the space we have the multiplication
49:45
map which maps the space into the space of multi-local functionals we define the n linear maps tn of the symmetric fox space to the space of multi-micro causal functionals just by
50:02
using the these bi-differential operator in terms of the feynman propagator and because we go to several factors we have to use the leibniz rule and this creates these indices ij here so this gives all the combinatoric feynman rules and so on and what is important is that this
50:28
multiplication map turns out to be bijective this is a little bit surprising because the the fox space you have functions of different field configurations and at the end your
50:42
functions of one field configuration but due to the smoothness of the local functionals so local functions are singular in on the diagonal but smooth along the diagonal so they're singular transversal to the diagonal but smooth along the diagonal and and this
51:06
then then allows to construct the inverse of m and at the end you can define this time ordering operator just by the direct sum of these maps tn composed with the inverse of m so this is a
51:24
definition of t and if you like path integrals you can also formulate this as a path integral namely it's just a convolution with the gaussian measure induced by the feynman propagator
51:42
but for the definition this does not help because you have to do the same calculations but the formula is there so you see the relation yeah so it's just a different way of formulating this now we can define the time ordered product namely the time ordered product
52:04
is now just equivalent to the classical product by the time ordering operator with this formula written there and so this is then this statement of this theorem which of course contains as an essential point that this multiplication map is invertible this was
52:22
done together with katashina reissner so particularly the time ordered product can be defined as a binary product of functionals and this product turns out to be associative and commutative because it's equivalent to the classical product then we can define the time
52:47
ordered exponential which is just the exponential series composed with this map t and by definition t is trivially on local functionals this means that we define our
53:03
functionals in the normal ordered form related to h so if you change h then we have also to change our functionals then we can define the interacting fields if we have some interaction l and this is uh something we might call the molar map from the freeze to the interacting theory
53:26
and this is just bogle huber's formula for the for the interacting field so we take the s matrix to the minus one product the time ordered product of the s matrix with the functional f and this is well defined if t to the minus one of f is in this set of
53:47
multi-local functionals yeah so in general there could be divergences but for this set it's well defined okay then uh as i said there's some ambiguity in the uh in the uh
54:04
definition of time order products and this ambiguity can be discussed in terms of the renormalization group in the sense of stuckelberg and peterman this is not exactly the same as the renormalization group of ursin so it's really a group it's a group of all
54:22
analytic bijections of the space of local functionals and the crucial relation is that this is additive if these functionals have disjoint support and the first derivative should be should be the identity and then there is something which was called the
54:42
main theorem of renormalization an unpublished paper by store and probably know and uh okay there are different versions of these theorems with different generality and uh the statement is that if i have found such a formal s matrix than
55:01
any other formal s matrix s hat is obtained from the original one by composing it with an element of this group so all questions of renormalization can then be deformed in properties of this group so for instance anomalies and so on can be discussed in terms of the group
55:24
okay now this is in the curved space everything is this is in the also in the curved space yeah this is a very general of course in flat space this is more or less what was known but but can be generalized to to curved space time now this but there remains one ugly problem
55:50
namely when one does this renormalization one has to do it for every point of the space time independently because there is no symmetry in general so how to compare these
56:06
renormalization procedures at different points of space time and of course there is some feeling how this should be done so if you want to have operation which is locally defined but to make this precise is rather complicated but uh of course this uh problem would not
56:25
does not exist on mikovsky space you have a sufficiently large symmetry you can have other space samples large symmetries where you have similar advantages but in the generic case you really have the problem that you that it seems that you have independent conditions at each
56:45
space time point which of course would be very ugly now the solution of this problem is to construct the theory not on one specific space time but on all space times in a coherent way
57:00
and this can again be formulated in the language of algebraic quantum field theory so we generalize the harkasler axioms in a useful way so the idea is that we look at sub regions as space times in their own right of course these sub regions should be
57:22
generally should be globally hyperbolic and the generalization is the following so we associate to every globally hyperbolic and okay contractible orientable and time-oriented manifold of a given dimension a certain c star algebra or in perturbation theory we
57:45
restrict ourselves to star algebras for every isometric causality at orientation preserving embedding we we have an injective homomorphism of the algebra of the region m to the algebra
58:05
of the space time n and this if you have two subsequent embeddings then this homomorphism just can be composed and then we have some condition of local commutativity so if these embeddings
58:25
map into space like separated sub regions and the corresponding algebras commute and there's dynamical law which just tells if the image of the space time m under chi contains a cushy surface
58:43
then a chi is even an isomorphism so in the language of category theories just tells you that the quantum field theory is the functor from the category of space times this admissible embeddings as morphisms to the category of unital star algebras with
59:05
injective homomorphisms as morphisms this is a concept which was called locally covariant quantum field theory and in a paper by together with Reiner Farsh and Romeo Brunetti and in this framework one can define a field independent of the space time
59:24
namely what is the field the field is just a natural transformation between the functor of test function spaces and the quantum field theory function functor so this means you have a field phi as a family
59:42
of maps phi m from the test function space of m to the algebra associated to m and which transforms in the appropriate way under these under these embeddings and so
01:00:00
If you use this more suggestive notation here, you see immediately what happens if n is equal to m, then chi, chi should preserve the matrix, so it's an isometry, so it's symmetry of the space time. This just is the usual notion of a covariant field on a given space time. So this condition contains the usual notion
01:00:22
of covariance, but it's valid in this more general framework. Now we can use this for renormalization. Namely, we define the time ordering operator as a natural transformation. And so this means if m is an admissible sub-region of n,
01:00:42
then the restriction of the time ordering operator to tm must coincide with the restriction of tn to the smaller sub-region must coincide with tm.
01:01:00
There's a practical obstruction to do this. There's a fact that there is no natural Hadamard function. This is related to this problem that there is no vacuum state. But this can be solved, this problem, and the solution was in a series of paper by Hollands and Ward.
01:01:20
What do you mean, it's solved? There is an ambiguity. No, no, the non-ambiguity is there, but you have to isolate the non-ambiguous. So the ambiguity is there, but the singularity, so if you look at the form of the Hadamard function, there are some parts which are non-ambiguous
01:01:42
and there's the ambiguous part. And you have to show that what you do does not depend on the ambiguous part. Actually, this is quite demanding. It's not an easy exercise. The Hadamard function means which kind of propagates? The Hadamard function just means
01:02:00
that you have a certain wave front set. But then I gave this explicit formula for the Hadamard function, which involved these smooth functions, u, v, and w. And it turns out that u and v are uniquely determined by geometry, and w is free. And you have to show, to say,
01:02:21
to discuss the dependence on w. That's what you have to do. Okay, up to now, I discuss this on the level of scalar theories. Now we can generalize it to Dirac, yeah? So removing these ambiguities, does that mean we can say what we mean
01:02:42
by when we say that the coupling constants of a quantum field theory are everywhere the same? Yeah, so if you have a dimensionless coupling constant, this is everywhere the same, yes. This would be the statement, yeah. Actually, usually there remains some ambiguities.
01:03:04
For instance, you have the coupling to the curvature, which can be tested, of course, only on a space with non-vanishing curvature. So if you start from Minkowski space, you cannot determine this constant, yeah? So you have to, but this is a finite number of parameters in a renormalizable field.
01:03:22
There also there would be a constant, sorry. Yeah, there would be a constant, again, yeah. Okay, so Dirac and Joanna fields can be discussed. This gives no fundamentally new problem, but you can make a lot of errors by making wrong signs and factors,
01:03:40
and I think it's difficult to find paper where all these signs are correct. But in gauge theories and gravels. I'm fine, I'm fine, I'm fine, I'm fine, I'm fine. Fine, I never consider the power of time. Yeah, okay, so in gauge theories and gravity, one has new problems because the Cauchy problem
01:04:02
is not well-posed due to the gauge theory. And in classical field theory, you would just fix the gauge. In quantum field theory, you cannot do this directly because these things which you would like to vanish have non-vanishing commutators or Poisson brackets
01:04:21
in the formalism of canonical field theory. But can you just pass the Dirac brackets then? Yeah, okay, the problem with Dirac brackets is that the singularity structures are very difficult.
01:04:44
So I think not any rigorous discussion of the Dirac brackets. So you say that in passing from Poisson to Dirac brackets, you eventually encounter a problem with singularities. The problem is, okay, I just don't know.
01:05:02
So I have not found any place where this was treated and, say, a sufficient generality. The problem is the following. All this in quantum field theory, at least in four-dimensional quantum field theory, you always have to smear over space and time. You cannot do smearing only in space.
01:05:20
This creates singularities. But the concept of Dirac brackets is usually defined at a fixed time. I mean, it's a really standard thing. For example, when we pass from Dirac to Majorana fermions that gives you just a factor of one half, and that's... Yeah, so for the linear fields themselves, this is okay.
01:05:42
The problem is if you go to non-linear fields because then you get additional singularities. And I just don't know whether this can be done. Okay, but in Yamil's series, we know how to do it using for the pop-up goals and anti-goals. And we have the BSD transformation.
01:06:03
And just at the end, we define the algebra of observables as a cohomology of the BRS operator. And we can try to do the same for gravity. But in gravity, we have a problem. Namely, if we do this just in the same way as in Yamil's,
01:06:20
we find that the cohomology is trivial because there are no local observables. This is a major problem. It can, in principle, be solved in the following way. Namely, we use the dynamical fields themselves as coordinates. This, of course, is not always possible.
01:06:40
This depends on the gravitational field. But for generic field configuration, this is possible. And so the idea is to... What do you mean, on-shell? Not necessarily on-shell. Just generically in the sense that there are no special symmetries and so on.
01:07:01
So that you have enough independent, say, curvature scalars, for instance. And then you just choose such a generally globally hyperbolic metric, and you expand then the extended action around this after second order. So you get a free field theory. One can show that this has Hadamard states.
01:07:25
And then one adds the remaining term of the action, the perturbation theory, and constructs everything. And this has to be done in such a way that the BST invariance holds. And at the end, you prove that if you change
01:07:42
the background infinitesimally, then the theory does not change. This is due to the principle of perturbative agreement proved by Hollons and Wald. And states can be constructed at the end by... If you start from a solution of the classical field
01:08:01
equation and use this concept of coherent states around this. Okay, so I come to the end. So I've tried to explain how this functorial approach to quantum field theory works. This was originally developed for the purposes
01:08:21
of renormalization on curved space time. But in principle, it allows also the framework for a background independent approach to quantum gravity. Of course, this does not change this problem of non-renormalizability, which means that you can perform everything step by step,
01:08:43
but you get new terms in the Lagrangian in every order. So in this sense, it's non-renormalizable, but I think it's probably possible,
01:09:04
nevertheless, to interpret this theory as a sense of effective field theory. At the end, actually, we have problems to see any effect of the quantization of gravity in experiments. So I think there's some hope
01:09:21
that these effects are very small. Actually, this is supported by work on the renormalization group in the group of Reuter and others. And at the moment, these formalisms are a little bit too far from each other, but principle,
01:09:40
I think this can be compared to this approach of Reuter. There's one major, major lack in this procedure. There's no application to physical phenomena. And this is due to one ugly feature,
01:10:00
namely the restriction to generic backgrounds. Namely, if you want to have an explicit example, you typically use not a generic background, but you use some of this background which has additional symmetries, and then you cannot directly use these concepts. So in this case, you have to use other methods.
01:10:22
So one way out would be to add certain fields, like, for instance, the dust fields of Brown and Kutcher, which have been used just for the same reason in loop quantum gravity. But this is work which has to be done, which was not done up to now.
01:10:40
Thank you very much for your attention. Yeah? I have a question. I mean, as far as I understand, your approach relies very much on Epstein-Klasser. Now, in quantum field theory, in practical quantum field theory, Epstein-Klasser is not used very much,
01:11:02
but people use dimensional realization. I mean, what would you advise, I mean, this is somehow related to the one but last point, if I actually want to do a practical calculation, how should I proceed? Yeah, so, okay.
01:11:24
So when you do it on Minkowski space, I think you can compare it only on Minkowski space, but on curved space time, there's not much, which was done in other formalism, actually, I think no complete proof. I think the only proof for curved space time is in the Epstein-Klasser framework.
01:11:43
But in the, on Minkowski space, Epstein-Klasser is just equivalent to other methods, so you use the method which is most useful for you. So there are certain cases where these Epstein-Klasser method leads to shorter calculations,
01:12:01
but there are also cases where it's more complicated. So I think I would, actually, what we analyzed was how to use dimensional regularization in the framework of Epstein-Klasser. This gives a position space version of the dimensional regularization, similar to what was already,
01:12:23
not really the same, but it's a spirit of some work of Bolini and Gambiaci. But you can also use other regularization methods, so that's not, yeah.
01:12:41
So I have two questions. I don't understand the last problem, but maybe that's too technical, I don't know. The other question is, is there any kind of Wilsonian version of the normalization group that they could use to analyze these field theories? Because naively it doesn't work,
01:13:02
because usually Wilson needs momentum space, but maybe there is a more clever. Yeah, so in principle you can, actually what you can do is you can replace, what I call this time ordering operator. You could replace this by some more regular object,
01:13:21
for instance, just by approximating the Feynman propagator by something regular, and then you get, you're just in this Porchinski version of the Wilson normalization group. Actually this has already been analyzed in a paper pass. So you really get the flow equation in this framework,
01:13:43
but you can use any approximation. So of course momentum space makes no sense, but you can use just something. Yeah, that's just, which I think might be a problem for the interpretation,
01:14:01
because you are free to use any, for instance you can just look at a sequence of test functions which converges in the sense of distribution to the Feynman propagator. And then you'll, yeah. Yeah, I mean it's a related question in a sense, because for me it's not abuse,
01:14:21
I think in general the normalization group, and Wilsonian normalization group will be related to the normalization group in the sense of normalization of the theory or the prediction of ambiguity, which are the free parameter, et cetera. So in your case what you define as a normalization group is clearly a group acting on ambiguities. Yes, yes. Is there any way you could see
01:14:42
that it has something to do with Wilsonian normalization group and change of scale? Yeah, you can see it, but the connection is not as close. So what you can do is you can,
01:15:01
you can discuss it in terms of these, okay, so what you can show is that in a certain sense the Wilsonian renormalization group
01:15:23
converges to the right thing. Yes, so you have, you use some regularization, you get your regularized time ordering operator,
01:15:43
you have your regularized formal S matrix, then you compute an element of the Stückelberg-Petermann renormalization group related to this transformation, and then you can show that in a certain sense these products converge.
01:16:01
But I think the statements which exist at the moment are not very strong. It's related to the question whether this perturbative calculation of the coefficients of the renormalization group gives the right answer for the Wilsonian renormalization group. I think that usually in the calculation
01:16:22
you usually neglect certain term which you consider to be irrelevant. And I think that's the question, how far this is justified. I think this answer has been given, to my knowledge, only in very special cases, particularly only for renormalizable theory. So that should be my, the relation only exists
01:16:41
for the mutable theory. There exists a relation, but I think the problem is the relation, as far as I know them, are not very strong. There's some similarity, but it's not exactly the same. Last question. In the case of quantum gravity, does your approach give a better definition
01:17:02
of what would be a quantum back reaction beyond your G0? You say G0 is off-shelf, but at some point you want a good G0 which incorporates back action. Yeah, yeah, yeah. Is there, does it help or not? Okay. Frankly. That's, of course, our hope. I think we are not so far.
01:17:20
So we hope the formalism in principle should give it, but when we, so what one should compare with the so-called semi-classical Einstein equation. The semi-classical Einstein equation has a problem that, a consistency problem. It's experimentally shown to be wrong, actually.
01:17:41
There is a proof experiment there. Okay, let me stop here because we want to go. Thank you.
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