Exploring String Theory on Curved Spacetime: How Does it Explain QFT?

[hellfire]

I was reading some of the first chapters of Zwiebach's book. It is explained how quantized relativistic strings turn out to describe an infinite set of possible particles and, therefore, that it is possible to consistently describe the results obtained in the Fock representation in QFT on a flat spacetime.

However, in QFT there are scenarios in which particles are not the fundamental entities. This is the case in curved spacetimes where particles depend on the chosen reference frame, because there is not always a global definition of the time coordinate that provides a unique set of positive frequency modes to build the Fock representation. Applications such as Hawking radiation or the particle creation in an expanding cosmological background are examples of this.

If I extrapolate what I have read so far about strings, it seams to me that the notion of string is also an observer dependent one. I am sure something is wrong with this view. Zwiebach shows how string theory can explain QFT in a flat background. How does it explain QFT on curved backgrounds? Before of moving on to more difficult references, I would like to get a feeling about this.

 

[Sauron]

Good question,i´ll try to think about it. But first i will add a question that it is somewhat related, i think.

In the Polyakov action you can get the interpretation of space coordinates of the string as fields in two deimensios.Precisely that is what allows the appearecnce of the superstring changing the bosonic fields which corresonde ot some coordinates by fermionic ones.

But,if we have fermionic fields and we come backto the geometric interpretation that means that some of the coordinates are in some way "fermionic coordinates".

But when doing geometric considerations it is always choosen standar Mikownskian N-Space (n usuallly 10, or elevenofrM-Theory).

That is, you have coordinats which corresponde to ferminoic fields but geometrically they are not specially distinguised. I have just reached these question and maybe it is an stupid one which i myself will answer in a few hours.Or may be not.

Of course if by some strange reason we need some "ferminic coordinates" (whatever it is), how does hawking effect apllies to it.

I dónt know about conmutative geometry nor string theory beyond the kaku book (i have readed something of a book by T banks on d-branes,but not very deeplly) so maybe the answer to my question is "noncomutative geometry".Well, tell me if so xD.

 

[Haelfix]

I believe when they are working with curved backgrounds (a hard problem), that they are required to foliate their spacetime with some choice of gauge, not unlike the technical procedures for working with qfts in curved spacetime.

 

[hellfire]

Thank you for your contributions. Sauron, you should elaborate a little bit more to make clear the relation of your comment to the question I have posed. Thanks.

 

[alexander1990a]

is background radiation independent on location?

 

[alexander1990a]

is background radiation independent of location?

 

[Sauron]

Thank you for your contributions. Sauron, you should elaborate a little bit more to make clear the relation of your comment to the question I have posed. Thanks.

Afther revising the relevant chapters in the aproppiate books i can

It was a lamentable missmemory of my side in fact. The thing is that for the superstring we have (local) supsersimetry in the world-sheet that afther GSO projection becomes target space supersimetry.

Well, let´s see how these relate somewhat with your question.

First of all let´s fix concepts. We have the polyakov action for the bosonic string. It describes the coupling of N (N=dimension of target space time) bosonics fields with world sheet gravity.

These means that we have N two dimensinal Klein-Gordon fields $$X_N(\sigma,\tau)$$ coupled to two dimesnional gravity in the world sheet described by a metric $$h_\nu_\mu$$.

Also we have a diferent interpretation for the $$X_N(\sigma,\tau)$$. They are, as I said, the coordinates of the embbeding of the world sheet in the N-dimensional target space. Their derviatives can be seen as the tangent vectors of the world-sheet surface of the string in the target space. And as tangent vectors they can be contracted with the target minkowsky metric.

Now we have two very diferent menings for teh derivatives of the $$X_N(\sigma,\tau)$$ fields. One meaining is the one which make them to be two dimensinal curved space Klein-Gordon fields and another meaning is that they are tangent vectors in the target space.

I said that the target space had a minkowskian metric. But in fact it can be a curved space. That would convert the Polyakov action in a quantum field theory known as a nonlinear sigma model. And yes, it would be necesary to think a bit (or more) about your questions.

But let´s go now to the superstring. Now in adition to the kelin gordon fields $$X_N(\sigma,\tau)$$ we have a whole two dimensional local super multiplete. That means fermion fields $$\Psi$$, matter (pseudo)scalars F, and also a graviton (the zweibein) a gravitino$$\chi$$ and an aditional A (pseudo)scalar.

That means that instead of a single $$X_N(\sigma,\tau)$$ K-G field which can be interpreted as a coordinate whose derivatives can be nterpreted as tangent vectors we have for every $$\mu$$ dimensional index five fields.

Sowe have that any dimension of space is not represented by a coordinate but by (at least) five coordinates (one for each kind of field).And it is not transparently clear that we have that the derivatives of that fields convert them in some kind of "tangent vectors" of that kind o strange space.

Well, I know supersimmetry but not the superspace formulation of it. Maybe we simply have that the superstring is "just simply" living in superspace (the anlogue of the "minkowsky" metric of the target space would be played here by the zweibein i guess).

Well,afther GSO projection the world-sheet supersimmetry becomes target space supersimetry. (local) Supersimmetry means we have a graviton and son that we have a curved target space time.

So we would have to take care of the Hawkin effect not in curved spacetime but in curved superspacetime (i.e. local supersimmetry, i.e. supergravity).

Or I think it could be so. I am not expert in string theory and maybe i just have missinterpreted the formulaes and their intuitive meaning (or simply i ignore some facts). But hey, we have stringy experts here which hopefully will clarify these viepoints ;).

P.S. Antoher thing which i have not clear is how care has ben taken of the times quesiton. We have a local parameter inthe world-sheet, $$\tau$$. And aditionally we have a time coordinate, t, in the target space. A target space which can be courved.And we know that by diffeormorfism invariance none of that times have an absolute mean. Has teh relation betwen these two, a priori different times relates? Are in fact, how some easy some argument that I ignore,the same?

 

[hellfire]

I do not understand everything you write there but I have some related questions that may be very trivial ones. My first question is about the meaning of $$X(\sigma,\tau)$$. Textbooks such as Zwiebach approach this topic explaining the motion of one single relativistic string in a fixed background, in which $$X(\sigma,\tau)$$ are the coordinates of the points of the world sheet.

Is this view correct in string theory? For example, as far as I know, string interactions arise from the same Polyakov action and there is no need to have a "second" $$X(\sigma,\tau)$$ in the action. Classically, if you are going to describe two strings you should make use of two different $$X(\sigma,\tau)$$ in the same action. Am I wrong? Does $$X(\sigma,\tau)$$ describe a single string?

Moreover, I am confused also by the fact that $$X(\sigma,\tau)$$ has a completely different status than the other parts of the action, or at least than the spacetime metric. I mean, in QFT there are only fields in the action. Here we have a field $$g_{\mu\nu}$$ covering all spacetime and another object $$X(\sigma,\tau)$$ with an unclear meaning for me.

 

[Sauron]

In favour of clarity expecify that $$X_N(\sigma,\tau)$$ should be $$\vec{X(\sigma},\tau)=(X_1(\sigma,\tau),X_2(\sigma,\tau),...,X_N(\sigma,\tau))$$

That is, $$\vec{ X}(\sigma,\tau)$$ has N components $$X_\mu(\sigma,\tau)$$.

But yes, once fixed that minor posiible notation source of confusion imust say you are righ.

We have two fields as you say, the external metric $$g_\mu_\nu$$ and
the string field $$\vec{ X}(\sigma,\tau)$$.

That is, the string doesn´t fix it´s own metric but instead you have an string propagating in an external background. The claims is that the graviton, which belongs to the particle spectrum of the string, would modify that background metric (it would be a perturbation to it). These unfair situation is claimed to be a provisiinal state on the string formalism and that future developments willmake a better task.

To be honest i am not up to date in string theory development and may be that future developments are already available somewhat (abut my guess is taht not).

The other thnig you say about needing a $$\vec{ X}(\sigma,\tau)$$ for any string inthe polyakov action is not exactly so.

The (bosonic) polyakov action strictly is a free particle action describing a single string (well, you can have a polyakov action for superstring,,but still describing a single superstring).

From that action you can fix gauge, obtain "particle" spectrums, do canonical quantization, whatever, for a single free string.

There is a related thing, the Polyakov integral. That is the analogous of the Feyman integral. It is these Polyakov integral which describes theinteraction betwen strings. As the name suggest it is based in the Polyakov action.

To describe the details of it is an unnecesry task because I believe you will have it in the zweibach book (i don't actually have that book, I have another string books--lust & theisen,kaku and others,but not zweibach-) but i don´t doubt yours will have a chapter (or many) in Polyakov action.

The key point is that as inany other perturbative theory the interaction gives you amplitudes betwen external states which behave acoording to the free theory.

If your feel unfair with strings not fixing it´s own background and another issues feel free to read Marcus and his labour in favour of LQG .

P.S. Another thing which you must consider is that the POlyakov actionis a first quantization action. That is, it is the analogous of the klein gordon action. There is not a $$\lambda\Phi^4$$ (i.e. interaction lagrangian) analoguous of the Polyakov action nor a second quantization "string field theory" (althought there are many attemps to construc onewicha are,as far as i know unsucessull).

Not having an interaction lagrangian in second quantization the Polyakov functional integral doesn´t follow the steps of the Feyman integral.

But for the hawkin process in fact you don´t need an interaction term. Only second quantized free incoming and outcoming fields. So maybe it is not necessary a second quantized interaction string but only a second quantized free string, which is easier to have i think, and for that you could purchase the same arguments as for puntual particles.

In fact i think there are many articles, even previous to the second string revolution, about first quantized strings in courved backgrounds.Hope somebody which knosws about it more than me would help us.

 

[hellfire]

OK, thank you, I see there is more stuff than I have assumed. I think I should go step by step, but nevertheless, I would like to ask whether there is a simple explanation you could provide me for the fact that there is no interaction term needed for the Polyakov path integral. This might make things a bit clearer.

 

[Sauron]

The key is to use vertex operators which select the in and out states and tomake conformal transformations to convert stringsintopointsand such that.

A.S.A.P as i get a few time a post a detailled answer.

 

[Sauron]

While thinkking about these i have got some doubts.

In the "intuitive" argument about the hawking effect it is told that a virtual pair is created and one of the particles of the pair is swallowed by the b-h.

But as i told the current formulation of string theory is a free theory. And it is an on-sheel theory also.

Even thought the idea behind a loop series is that you have virtual particles (in these case virtual strings,or virtual surfaces) appearing. But as there is no a string field to go with it i don´t know how much sense it makes to talk about virtual string modes at all.

Of course i have readed the technical explanation of the hawking effect (also i always loose mysellf somewhat with the conformal-Penrose diagramsn going on) and i don´t see a clever relation with that vision and the need of virtual particles so maybe you just need asyntotic string states.

B.T.W. the idea is that you make conformal transofrmations (in the sense of complex variable theory) of the string world sheet.

The conformal maping is made such that you always map an asyntotic string state to a point in a sphere (an sphere with n-handles if you have an n-loop string amplitude). That aysntothic state is maped by the conformal transformation into the vertex operator. (you can do these because the conformal invariance of string theory, as it is easilly seen as a remaining symmmetry once you cfix the action to the so called conformal gauge assures the cross sections are conformal invariant).

The path integral integrates over all the posible positions of the vertex operator in the sphere

B.T.W. My previous information about the GSO projection is necesary for the NS-R (Neveu-Schwartz-Ramond) action.You also can have the GS (green-schwartz) action which is manifestly supersymmetric in the target space.But it is highly non linear and difficoult to quantice,even for the free theory).

 

[Demystifier]

However, in QFT there are scenarios in which particles are not the fundamental entities. This is the case in curved spacetimes where particles depend on the chosen reference frame, because there is not always a global definition of the time coordinate that provides a unique set of positive frequency modes to build the Fock representation. Applications such as Hawking radiation or the particle creation in an expanding cosmological background are examples of this.

If I extrapolate what I have read so far about strings, it seams to me that the notion of string is also an observer dependent one. I am sure something is wrong with this view.

In field/particle physics, there is a question whether the fundamental entities are fields or particles. If the fundamental entities are fields, then particles may be observer dependent. But if fundamental objects are particles (and this possibility should not be excluded) then it is the standard view of QFT in curved spacetime that should be modified.

The situation in string theory is completely analogous. Are the fundamental objects strings or string fields? We do not know, but the fact is that we do not understand string field theory very well. Therefore, I would say that the view that "particles are more fundamental objects than fields" seems more viable in string theory than in the standard elementary-particle theory.