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TRANSCRIPT
From Quantum Field Theory to Strings
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½÷(H. Lu)®Æ
@ICTS-USTC
4:00pm, September 30, 2016
• Introduction: the Fundamental Nature of Matter
• Free Quantum Field Theory
• Interacting Quantum Field Theory
• Gauge Theory
• String Theory
• Matter and spacetime
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Questions asked by physics
• What: basic constituents
• How: their interactions
• Whyµunderlying symmetry principle
Interaction: In Newtonian physics, this is the concept of “force”.The Newtonian physics, however, cannot explain why 1/r2-lawfor gravity or electrostatic forces.
Symmetry: It is about “beauty”. Before Galleon, symmetry con-sidered is typically discrete, except for the continuous symmetryof a sphere. Kepler tried to use platonic solids to build a modelfor the Solar system.
Discrete symmetry may be able to explain some kinematics, butvery difficult, if not impossible, to explain dynamics, since dy-namics, as far as we know, is governed by differentiations.
Our focus: the nature of matter
Regardless the metaphysical discussion on matter, the most im-portant dispute is: Is matter continuous or discrete? (e.g. light.)
Owing to the observed continuous nature of space, it is naturalto believe matter is also continuous; otherwise, matter can onlybe made of particles that are point-like, which is an impossibleconcept in the classical physics. It leads to singularity at shortdistance.
The concept of continuous matter is also very hard to imaginesince it is difficult to explain how the infinitesimal quantities canhold together.
Some ancient Greeks proposed that matter is composed fromquantities called indivisible and indestructible atoms (Leucippus,Democritus.) In order to explain the variety of matter, they wouldbelieve a vast number of different atoms.
Newtonian physics instinctively treats matter as point-like, al-though, owing to calculus, it can handle also continuous matter.
Object-like matter: atoms
The idea of atom became a scientific concept since the firstevidence-based theory was proposed by in the early 1800s. (JohnDalton)
With the discovery of fundamental particles such as electron, itbecomes increasingly natural to adopt the idea that the appar-ently continuous matter consists of discrete point-like particles.(e.g. water, air, etc.)
It leads following conclusions: (1) The particle number must beconserved. (2) If one-type of particles can change to anothertype, there must be internal structures and hence not fundamen-tal and point-like.
It becomes also impossible to explain the phenomenon of identicalparticles: any two electrons are identical besides their positionsand velocities.
Another type of matter: EM and photon
Electric-magnetic field are continuous and their ripples or wavesgive rise to light.
In high school physics class, it is strongly emphasized that EMfield is physical matter, rather than a mathematical construct.This emphasis is necessary because they are different from othermatter: they do not appear to be a collection of fundamentalpoint-like particles.
Thus classically, we now have two very different types of matter:particle-like (e.g. electron) and wave-like (e.g. light) that areripples of fields.
The wave-particle duality on the other hand tells that the prop-erties of electrons and photons are fundamentally similar.
If photon is the quantized EM field, shouldn’t an electron be thequantized “electron field”?
Should not one introduce the concept of electron field?
Particle vs. field
Continuous matter in Newtonian physics is viewed as an approx-imation of a collection of large number of particles:
mid2~xidt2
= ~Fi = −~∇iV , V = (~x1, ~x2, . . . , ~xn) .
Here i = 1,2,3, . . . , n is the label of different particles and ~xi’sare the coordinates of the particles. In this picture, the coor-dinates ~xi = ~xi(t) are dynamical variables, whilst i is the labeldistinguishing different particles.
A field is a quantity defined at every point of space and time(~x, t), with the general expression
φa(~x, t) , labels : (a, ~x) .
• Particle mechanics: position, or coordinates are dynamicalvariables.
• Field theory: coordinates are merely labels and not dynamical.~x 6= ~x(t), instead ~x and t are in the similar footing. (This isconsistent with special relativity.) This leads to an infinitedegrees of freedom: at least one at each point ~x in space.
• The label a can be spacetime direction, spinor or group in-dices, etc.
Field vs. QM wave function
Wave function in quantum mechanics ψ(~x, t) resembles a field(which leads to so-called second quantization;) however, thereare significant differences:
• A field is a classical concept that can be measured directly.A wave function in QM does not have a classical limit andcannot be measured directly.
• A field contains infinite degrees of freedom, and a wave func-tion ψ(~x, t) describes only a single particle. The wave functionfor multiple particles takes the form ψ(~x1, ~x2, . . . , ~xn; t).
• At the quantum level, a field is promoted to become on op-erator, whilst a wave function is a function of subset of com-muting operators and hence can be c-numbers.
Quantum field theory
Quantum field theory regards fields as fundamental, and particlesare quantum excitations of the fields.
However, there is then a question:
Why do we need not to introduce “electron field” in classicaltheory of electrons, as in the case of the electromagnetic fieldfor photons?
Relativistic quantum field theory
In this lecture, quantum field theory refers to a quantum theoryof relativistic field theory.
A key feature of a relativistic theory is that, in addition to the timeand space translational symmetry, it is further invariant under theLorentz symmetry of spacetime:
x′µ = Λµνxν , (Λµν constant)
that leaves the Minkowski metric
ds2 = dt2 − dx2 − dy2 − dz2 = ηµνdxµdxν .
In other words,
ηµνΛµρΛνσ = ηρσ , or ΛTηΛ = η .
The Lorentz group is SO(1,3), a Lorentzian generalization of theSO(4) rotational group of the Euclidean 4-space.
Note: Translational transformation, xµ → xµ + aµ where aµ areconstants, is not linear from the group theoretical point of view.
Representation of Lorentz group
To build a field theory that is invariant under Lorentz group,its field must belong to a representation of the Lorentz group.Its characteristics can be specified by its spin, which must beintegers or half integers.
• spin-0: scalar φ(x), invariant under the Lorentz transforma-tion, also called singlet. From now on, x denotes (t, ~x),
• spin-12: (Dirac) spinor ψα(x), α = 1,2,3,4 spinor indices,
• spin-1: Vector Aµ(x), e.g. Maxwell field,
• spin-32: ψαµ(x),
• spin-2: hµν(x) with hµν = hνµ
• ...
In quantum field theory, typically, one deals with particles of spins(0, 1
2,1). The higher spins arise in QFT usually from compositefields.
Equations of motion
A key property of Einstein’s special relativity is that E = Mc2,which is derived from (setting c = 1)
E2 = ~p2 +m2 .
According to the wave-particle duality: E → i∂t and ~p = −i~∂.Thus any field φa must satisfy
(+m2)φa = 0 , ≡ ∂µ∂µ = ∂2t − ∂2
x − ∂2y − ∂2
z .
Here φa can be a scalar, a spinor or a vector, etc.
Real scalar field
A free scalar field φ(x) satisfies the Klein-Gorden equation
(+m2)φ = 0 ,
It can be derived from the action S
S =∫dtL =
∫dtd3xL =
∫d4xL ,
L =1
2∂µφ∂
µφ−m2φ2
=1
2(∂φ)2 −m2φ2 , (∂φ)2 ≡ ∂µφ∂µφ .
L: Lagrangian, L: Lagrangian density, sometimes called also asLagrangian for simplicity.
Euler-Lagrange equation
For a general Lagrangian density
L = L(φa, ∂µφa) .
The equations of motion are given by the Euler-Lagrange equa-tion:
∂µ
(∂L
∂(∂µφa)
)− ∂L∂φa
= 0 .
Conserved quantities
Noether theorem tells us that a continuous global symmetry im-plies a conserved charge.
(1) Spacetime translational symmetry: Conservation of energy-momentum tensor
Tµν = ∂µφ∂νφ−1
2ηµν(∂φ)2 +
1
2m2φ2ηµν .
Tµν = T νµ and ∂µTµν = 0. Conserved 4-vector
Pµ =∫
t=consd3xTµ0dΣ0 .
Note that T00 turns out to be the Hamiltonian. (Come to thislater.)
(2) Lorentz symmetry: Conserved current
(Jµ)ρσ = xρTµσ − xσTµρ , ∂µ(Jµ)ρσ = 0 .
(3+1)-dimensional generalization of the three angular momentaL1 = x2p3 − x3p2, etc.
Hamiltonian formalism
In the canonical quantization in quantum mechanics, one needsto know the Hamiltonian of the system. We start by defining themomentum πa(x) conjugate to φa(x)
πa(x) =∂L∂φa
.
The Hamiltonian density is given by
H(φa, πa) = πa(x) φa(x)− L .
The Hamiltonian is given by
H =∫d3xH(φa, π
a) .
For the real free scalar, we have
H = T00 =1
2φ2 +
1
2(∇φ)2 +
1
2m2φ2 .
Note that it is non-negative, and hence there is no negative-energy problem.
(This is not true if one treats φ as the QM wave function, asSchrodinger tried and failed in his early attempts to constructrelativistic quantum mechanics.)
Complex scalar
Let us now consider the Lagrangian for the two real scalars φ1and φ2 but with the same mass
L =1
2(∂φ1)2 +
1
2(∂φ2)2 − 1
2m2(φ2
1 + φ22) .
It is economical to define a complex scalar field
Φ =1√2
(φ1 + iφ2) , Φ∗ =1√2
(φ1 − iφ2) .
The Lagrangian density can be expressed as
L = ∂µΦ∗∂µΦ−m2 Φ∗Φ .
In addition to the spacetime symmetry, there is an additionalglobal symmetry
Φ→ eiαΦ , Φ∗ → e−iαΦ∗ .The corresponding conserved current is
jµ = i((∂µΦ∗)Φ−Φ∗∂µΦ
).
Quantizations: Different Pictures
• Schrodinger picture: state vectors evolve in time, but opera-tors are constant with respect to time.
• Heisenberg picture: state vectors are constant with respectto time, but physical operators evolve in time.
• (Dirac) interacting picture.
Canonical Quantization: Particle vs. Field
Canonical quantization: Hamiltonian based, with the generalizedcoordinates qa and their conjugate momentum pa promoted tobecome quantum operators. The corresponding Poisson bracketsbecome commutators
[qa, qb] = 0 = [pa, pb] , [qa, pb] = i δba .
where we are using natural units so that ~ = 1.
In field theory, we do the same for the field φa and its momentumconjugate πa, and promote them to physical quantum operators.We shall work in Schrodinger picture with the operator indepen-dent of the time, so that φa = φa(~x) and πa = πa(~x), obeying
[φa(~x), φb(~y)] = 0 = [φa(~x), φb(~y)] , [φa(~x), πb(~y)] = iδ3(~x− ~y)δba .
Free fields vs. harmonic oscillators
Let us consider a free scalar field with
L =1
2(∂φ)2 − 1
2m2φ2 .
The equation of motion is ( + m2)φ = 0. Make a Fouriertransformation
φ(~x, t) =∫
d3p
(2π)3ei~p·~xφ(~p, t) .
The Klein-Gordon equation becomes
(∂2t + ~p2 +m2)φ(~p, t) = 0 , → (∂2
t + ω2)φ = 0 ,
This is precisely the equation for a simple harmonic oscillatorwith frequency
ω~p = +√~p2 +m2 .
Thus the most general solution to the Klein-Gordn equation is alinear superposition of simple harmonic oscillators, each vibratingwith a different frequency and with different amplitudes.
Quantizing φ(~x, t) amounts to quantizing this infinite number ofsimple harmonic oscillators.
Review of a simple harmonic oscillator
Consider H = 12p
2 + 12ωq
2, with the quantum mechanical commu-tation relation [q, p] = i. Define
a =√ω
2q +
i√2ωp , a† =
√ω
2q − i√
2ωp .
we have
[a, a+] = 1 , → H =1
2ω(aa†+ a†a) =
1
2ω(a†a+
1
2) .
Since [a, a†] = 1, the vacuum |0〉 cannot be annihilated by botha and a†, but it is consistent to define
a|0〉 = 0 .
The physical states are then quantized, with
|n〉 = (a+)n|0〉 , H|n〉 = (n+1
2)ω|0〉 .
Quantizing a free real scalar field
In Schrodinger picture:
φ(~x) =∫
d3p
(2π)3
1√
2ω~p[a~p e
i~p·~x + a†~p e−i~p·~x] ,
π(~x) =∫
d3p
(2π)3(−i)
√ω~p
2[a~p e
i~p·~x − a†~p e−i~p·~x] ,
with ω~p = +√~p2 +m2. The commutation relations
[φ(~x), φ(~y)] = 0 = [π(~x), π(~y)] , [φ(~x), π(~y)] = iδ(3)(~x− ~y) .
become
[a~p, a~q] = 0 = [a†~p, a†~q] , [a~p, a
†~q] = (2π)3δ(3)(~p− ~q) .
Thus
H =1
2
∫d3x(π2 + (∇φ)2 +m2φ2)
=1
2
∫d3p
(2π)3ω~p(a~pa
†~p + a
†~pa~p)
=1
2
∫d3p
(2π)3ω~p[a
†~pa~p +
1
2(2π)3δ(3)(0)] .
Dealing with vacuum energy
Defining vacuum
a~p|0〉 = 0 , ∀~p .Then we find
H|0〉 =∫d3p
1
2ω~p δ
(3)(0)|0〉 =∞|0〉 ≡ E0|0〉 .
(2π)3δ(3)(0) = limL→∞
∫ L2
−L2d3xei~p·~x|~p=0 = lim
L→∞
∫ L2
−L2d3x = V .
Divergence at large volume: Infra-red (IR) divergence. It is thusmore natural to compute energy-density
ε0 =E0
V=
∫d3p1
2 ω~p δ(3)(0)
(2π)3δ(3)(0)=∫
d3p
(2π)3
1
2ω~p .
It is now divergent as |~p| → ∞, and is known as an ultra-violet(UV) divergence.
Normal ordering
What is physically important is not the absolute energy, but theenergy difference. One can apply a uniform rule such that allthe annihilation operators placed to the right, to get rid of thevacuum divergence. Thus
:aa† := a†a , :a†a := a†a .
:H :=∫
d3p
(2π)3ω~p a
†~p a~p , → :H : |0〉 = 0 .
Note that when the theory couples to gravity, the absolute ener-gy will have a physical meaning, giving rise to the cosmologicalconstant.
From fields to particles
The normal-ordered Hamiltonian
H =∫
d3q
(2π)3ω~q a
†~q a~q .
It can be shown
[H, a†~p] = ω~p a†~p , [H, a~p] = −ω~p a~p .
As the harmonic oscillator, we can construct energy eigenstate|~p〉 = a
†~p|0〉:
H|~p〉 = ω~p |~p〉 .Momentum and angular momentum:
~P =∫
d3p
(2π)3~p a†~pa~p , → ~P |~p〉 = ~p |~p〉 .
Ji = −iεijk
∫d3p
(2π)3a†~p
(pj
∂
∂pk− pk ∂
∂pj
)a~p .
It can be shown that Ji|~p = 0〉 = 0, implying a scalar field givesrise to a spin-0 particle, a particle with no internal angular mo-mentum.
Multi-particle states and identical particles
Particles, as excited states of a given field, are necessarily identi-cal. For example, if there is only one electromagnetic field in ouruniverse, it is then necessary that all the photons are identical.
This is reason why although there are about 1080 particles in theobservable universe, the standard model for all these particles canbe written in a T-shirt with big font.
A two-particle state of momenta ~p1 and ~p2:
|~p1, ~p2〉 = a†~p1a†~p2|0〉 .
∵ [a†~p1, a†~p2
] = 0 ,∴ |~p2, ~p1〉 = a†~p2a†~p1|0〉 = a
†~p1a†~p2|0〉 = |~p1, ~p2〉 .
This describes a pair of identical particles with Bose statistics.
An n-particle state
|~p1, ~p2 . . . ~pn〉 = a†~p1a†~p2. . . a
†~pn|0〉 .
Quantizing a complex scalar
As with the real scalar field, we expand ψ as a sum of plane waves
Φ =∫
d3p
(2π)3
1√2E~p
(b~p e
i~p·~x + c†~p e−i~p·~x) ,
Φ† =∫
d3p
(2π)3
1√2E~p
(b†~p e−i~p·~x + c~p e
i~p·~x) ,
The conjugate canonical momentum π = Φ† is
π =∫
d3p
(2π)3i
√E~p
2
(b†~p e−i~p·~x − c~p ei~p·~x) ,
π† =∫
d3p
(2π)3(−i)
√E~p
2
(b~p e
i~p·~x − c†~p e−i~p·~x) .
Note that b~p and c†~p are independent now because Φ is complex.
[b~p, b†~q] = (2π)3δ(3)(~p− ~q) = [c~p, c
†~q] .
Particle and anti-particle
Energy and conserved global charge
H =∫
d3p
(2π)3ω~p(b†~p b~p + c
†~p c~p
).
Q = i∫d3(πΦ−Φ†π†) =
∫d3p
(2π)3
(b†~p b~p − c
†~p c~p
).
It is clear
[H, b†~p] = ω~pb†~p, [H, b~p] = −ω~pb~p, [Q, b†~p] = b
†~p, [Q, b~p] = −b~p,
[H, c†~p] = ω~pc†~p, [H, c~p] = −ω~pc~p, [Q, c†~p] = −c†~p, [Q, c~p] = c~p,
The (b†, b) are the creation and annihilation operators for par-ticles, whilst (c†, c) are creation and annihilation operators foranti-particles.
Q = Nb −Nc is the “total” particle numbers that are conserved.When the global symmetry Φ→ eiαΦ is gauged, Φ becomes thecharged particle, and Q is the total charge and it is conserved.
A real scalar particle is its own anti-particle.
From QFT to QM
Leave this to an excise.
Question: If we interpreted the scalar φ(x) in the Klein-Gordonequation as quantum wave function, it suffers from having neg-ative energy states, since
E±~p = ±√~p2 +m2 .
How is this problem solved in QFT?
Maxwell field: spin-1
L = −1
4FµνF
µν , Fµν = ∂µAν − ∂νAµ .
eom : ∇µFµν = 0 , Bianchi identity: ∂[µFνρ] = 0 .
Gauge symmetry: The Lagrangian is invariant under the gaugesymmetry
Aµ → Aµ + ∂µΛ .
Fµν =
0 −E1 −E2 −E3E1 0 B3 −B2E2 −B3 0 B1E3 B2 −B1 0
,
Fi0 = −F0i = Ei , Fij = εijkBk .
Note that Aµ = (φ, ~A).Exercise: Recover the familiar the Maxwell equations.
Dirac Fermions: spin-12
The Dirac equation in the form originally proposed by Dirac is(βmc2 + c(α1p1 + α2p2 + α3p3)
)ψ(x, t) = i~
∂ψ(x, t)
∂t,
where αi and β are 4× 4 matrices satisfying
α2i = β2 = I4 , αiαj + αjαi = 0 , αiβ + βαi = 0 .
They are anti-commuting with each other.
It is superficially similar to the Schrodinger equation of a freeparticle:
− ~2
2m∇2φ = i~
∂
∂tφ ,
which is a translation of the Newtonian energy E = p2
2m to thelanguage of operators.
The special relativity (SR) of Einstein tells us
E2
c2= ~p2 +m2c2 .
This gives rise to the Klein-Gordon equation(∇2 − 1
c2∂2
∂t2
)φ =
m2c2
~2φ .
This is the case we have discussed and we shall not consider itfurther.
Dirac thus thought to try an relativistic equation that was firstorder in time . One could, for example, formally take the rela-tivistic expression for the energy
E = ±√~p2 +m2 ,
replace p by its operator equivalent, expand the square root inan infinite series of derivative operators, set up an eigenvalueproblem, then solve the equation formally by iterations. Mostphysicists had little faith in such a process, even if it were tech-nically possible.
Before we tells the story what Dirac did, let us consider a simplerproblem, with only 1 spatial dimension:
−2 =∂2
∂x2− ∂2
∂t2.
We want to express 2 = −L2 where L is a first-order differentialoperator:
L = a∂x + i b∂t .
Thus
L2 = LL = (a∂x + i b∂t)(a∂x + i b∂t)= a2∂2
x − b2∂2t + ia, b∂x∂t ,
wherea, b ≡ ab+ ba .
Thus if we can find a and b such that
a2 = 1 = b2 and a, b = 0 ,
we shall have
−2 = L2 .
We can then turn the Klein-Gordon equation φ = m2φ to
iLψ = mψ .
Quaternions¶¶¶ Clifford algebra
Obviously, there is no real or complex numbers that can satisfythe condition a, b = 0. However, quaternions do:
i2 = j2 = k2 = ijk = −1 .
Quaternions (1, i, j, k) can be realized by 2 × 2 Pauli matrices,namely
i = σ1 , j = σ2 , k = σ3 ,
together with I2, the identity matrix.
We can now choose, say
a = σ1 , b = σ2 .
Thus we see that ψ has two components.
Now back to the Dirac story, which tells that Dirac was staringinto the fireplace at Cambridge, pondering this problem, when hehit upon the idea of taking the square root of the wave operatorthus:
∇2 − ∂2
∂t2= L.L ,
L = A∂x +B∂y + C∂z + iD∂t ≡ γµ∂µ .
The above equation can work provided that
γµ, γν = 2ηµν .
The simplest solution to the γ matrices has to be 4× 4, e.g.
γ0 =
(0 I2I2 0
), γ1 =
(0 σx−σx 0
),
γ2 =
(0 σy−σy 0
), γ3 =
(0 σz−σz 0
).
Note that we can introduce
γ5 = iγ0γ1γ2γ3 =
(I2 00 −I2
).
We have γ5, γµ = 0. Note that γM = γµ, γ5 can be viewed as
γ-matrices in D = 5, with
ΓM ,ΓN = 2ηMN .
Dirac equations
The Dirac equation
(iγµ∂µ −m)ψ = 0 , i∂µψγµ +mψ = 0 .
Act on the Dirac equation with (iγν∂ν +m):
0 = (iγν∂ν +m)(iγµ∂µ −m)ψ = −(γνγµ∂ν∂µ +m2)ψ .
∵ γνγµ∂µ∂ν =1
2γν, γµ∂ν∂µ = ∂µ∂
µ = ,
∴ (+m2)ψ = 0 , Klein-Gordon equation
We recover the Klein-Gordon equation. Thus the Dirac fermionmust satisfy the Klein-Gordon equation. The Lagrangian density
L = ψ(iγµ∂µ −m)ψ .
Dirac fermion has four components, and the quantization givesrise to an electron and its anti-particle positron, with each hasspin up and down.
Interacting theory
So far we have considered free theories of fields, whose La-grangian is characterised as being bilinear in terms of the fields.What about interactions?
In Newtonian theory, an interaction is introduced via a force, ora potential of coordinates. This will not work in QFT, sincecoordinates are only labels, not dynamical. In QFT, interactionsmanifest themselves as higher-order terms in the Lagrangian.
For example, in the real scalar theory, we can have
L =1
2(∂φ)2 − V (φ) , V = m2φ2 + λφ4 .
(Note that if m2 < 0, corresponding to tackyon field, the vac-uum φ = 0 is not stable, giving rise to spontaneous symmetrybreaking.)
Interaction via gauging: QED
Let us consider the Dirac fermion for electrons:
L = ψ(iγµ∂µ −m)ψ .
It is invariant under the global symmetry
ψ → eiαψ .
By gauging, we mean that we let the constant α be dependentnow on xµ. The Lagrangian then is no longer invariant. Theculprit that causes the non-invariance is ∂µ. We can add a con-nection Aµ, namely
∂µ =⇒ Dµ ≡ ∂µ − ieAµ ,
such that under ψ → eiαψ, we have Aµ → Aµ + 1/e ∂µα. Thisensure that Dµψ transforms like ψ, namely
Dµψ → eiαψ .
The Lagrangian is then invariant under the gauge symmetry.
But this cannot be all, since we do not have the kinetic termfor Aµ. The transformation implies that Aµ is nothing but theMaxwell field, and hence we have
Quantum electrodynamics
L = −1
4FµνFµν + ψ(iγµDµ −m)ψ ,
= −1
4FµνFµν + ψ(iγµ∂µ −m)ψ + eAµψγ
µψ .
The Maxwell equation now becomes
∂µFµν = jν ≡ ψγνψ .
where jµ is the electric 4-vector current.
This is the Lagrangian for QED. The interaction term, namelythe last term, arises from gauging the global U(1) symmetry.
We start with a linear theory with no interaction, but the processof gauging the global symmetry leads to an interacting termAµψγµψ.
Interestingly, this is how the real world operates: The interactionultimately gives rise to the Coulomb’s law.
Charged complex scalar
Complex scalar can also be charged. We may start from
L = ∂µΦ∗∂µΦ− V (Φ∗Φ) .
It is also invariant under the global symmetry Φ → eiαΦ. Theanalogous gauging leads to
L = −1
4FµνFµν +DµΦ∗DµΦ− V (Φ∗Φ) .
Generalizing U(1) ∼ SO(2)
In QFT, there are two types of fields: The Dirac-type fermionshave to be introduced by hand; The Maxwell-type of fields canbe derived from gauging global symmetries. The complex scalarcan be expressed as two real scalars:
Φ =1√2
(φ1 + iφ2) ,
The Lagrangian is
L =1
2(∂φ1)2 +
1
2(∂φ2)2 − V (φ2
1 + φ22) .
The U(1) transformation Φ → eiαΦ can be expressed as SO(2)rotation between φ1 and φ2:
(φ1φ2
)→(
cosα − sinαsinα cosα
)(φ1φ2
).
Yang-Mills (Nonabelian) from gauging
Now let us imagine three real scalars of the same mass, with theLagrangian
L =1
2(∂φ1)2 +
1
2(∂φ2)2 +
1
2(∂φ2)2 − V (φ2
1 + φ22 + φ2
3) .
It is invariant underφ1φ2φ3
→M
φ1φ2φ3
, MTM = I3 .
This is an SO(3) group. Now let the SO(3) element M be xµ
dependent, we must introduce connection to ∂µφi such that
∂µφi → (Dµφ)i = ∂µφ
i − g εijkAjφk ,where Ai’s are the gauge potentials for the SO(3) ∼ SU(2) Yang-Mills fields.
QCD (of single flavour)
The Lagrangian is
L = −1
4F aµνF aµν + ψa
(iγµ(Dµψ)a −mψa
).
Here a is the SU(3) color indices, and ψa’s are the quarks andAa’s are the gluons.
Strangely enough, the world at the fundamental level follows theabove rather simple and very mathematical construction.
From interaction to Newtonian force.
The interacting term in the QED we discussed earlier is
Lint = Aµψγµψ .
This describes a three-point interaction between an electron, apositron and a photon. This is the basic interaction in QED.This leads to the Coulomb’s law. To see this, let us mimic thisinteraction by considering a simpler scalar-Yukawa theory
L = L0 + Lint ,
L0 = ∂µΦ∗∂µΦ +1
2(∂φ)2 −M2Φ∗Φ− 1
2m2φ2 ,
Lint = −gΦ∗Φφ .Φ and Φ∗ describe nucleon and anti-nucleon; φ describes meson.
Three pictures of quantum mechanics
Schrodinger picture: Operators Os are fixed and states evolvesas
id|ψ〉sdt
= H|ψ〉s .
Heisenberg picture: states are fixed and operators evolve as
dOHdt
= i[H,OH] .
We can map from one picture to the other as follows
OH(t) = eiHtOse−iHt , |ψ〉H = eiHt|ψ〉s .
Interacting picture: Consider H = H0 +Hint. We require that
• Time dependence of operators is governed by H0:OI = eiH0tOse−iH0t;
• Time dependence of states is governed by Hint:
id|ψ〉Idt = HI |ψ〉I.
State evolution
The states evolve by an unitary operator U such that
|ψ(t)〉I = U(t, t0)|ψ(t0)〉I .The solution for U is given by
U(t, t0) = T [exp(−i∫ tt0HI(t
′)dt′)] ,
where T stands for time-ordering with the following rule
T (O1(t1)O2(t2)) =
O1(t1)O2(t2), if t1 > t2O2(t2)O1(t1), if t2 > t1
Scattering amplitudes
Consider the scalar Yukawa theory with
Hint = g∫d3xφΦ†Φ .
• φ = a+ a†, creating or destroying meson particle.
• Φ = b+ c†, creating anti-nucleon and destroying nucleon.
• Φ = b†+ c, creating nucleon, and destroying anti-nucleon.
Meson decay: φ→ Φ†ΦNucleon scattering: Φ†Φ→ φ→ Φ†Φ.
The amplitude to go from |i〉 to 〈f | is given by
limt±→±∞
〈f |U(t+, t−)|i〉 ≡ 〈f |S|i〉 .
The unitary operator S is known as the S-matrix, i.e., a matrixdescribing scattering amplitudes.
Meson decay φ→ Φ†Φ
|i〉 =√
2E~p a†p|0〉 , |f〉 =
√4E~q1
E~q2b†~q1c†~q2|0〉 .
At the tree level (lowest order)
〈f |S|i〉 = −i g (2π)3δ(4)(q1 + q2 − p) .
A particle with no internal structure can decay to other particles!
Feynman diagram illustration:
solid lines to denote its charge; we’ll choose an incoming (outgoing) arrow in the
initial state for ψ (ψ). We choose the reverse convention for the final state, where
an outgoing arrow denotes ψ.
• Join the external lines together with trivalent vertices
ψ
ψ+
φ
Each such diagram you can draw is in 1-1 correspondence with the terms in the
expansion of 〈f |S − 1 |i〉.
3.4.1 Feynman Rules
To each diagram we associate a number, using the Feynman rules
• Add a momentum k to each internal line
• To each vertex, write down a factor of
(−ig) (2π)4 δ(4)(∑
i
ki) (3.54)
where∑ki is the sum of all momenta flowing into the vertex.
• For each internal dotted line, corresponding to a φ particle with momentum k,
we write down a factor of∫
d4k
(2π)4i
k2 −m2 + iǫ(3.55)
We include the same factor for solid internal ψ lines, with m replaced by the
nucleon mass M .
– 61 –
Nucleon scattering
|i〉 =√
4E~p1E~p2
b†~p1b†~p2|0〉 ≡ |p1, p2〉 .
|f〉 =√
4E~q1E~q2
b†~q1b†~q2|0〉 = |q1, q2〉 .
The result is
〈f |S − 1|i〉 = i(−ig)2[
1
(p1 − q1)2 −m2+
1
(p1 − q2)2 −m2
].
The results can be illustrated using Feynman diagram:
3.5 Examples of Scattering Amplitudes
Let’s apply the Feynman rules to compute the amplitudes for various processes. We
start with something familiar:
Nucleon Scattering Revisited
Let’s look at how this works for the ψψ → ψψ scattering at order g2. We can write
down the two simplest diagrams contributing to this process. They are shown in Figure
9.
p2
p1
p1/
p2/
p2
p1
/
p/
2
1
+
p
k k
Figure 9: The two lowest order Feynman diagrams for nucleon scattering.
Applying the Feynman rules to these diagrams, we get
i(−ig)2[
1
(p1 − p ′1)
2 −m2+
1
(p1 − p ′2)
2 −m2
](2π)4 δ(4)(p1 + p2 − p′1 − p′2) (3.56)
which agrees with the calculation (3.51) that we performed earlier. There is a nice
physical interpretation of these diagrams. We talk, rather loosely, of the nucleons
exchanging a meson which, in the first diagram, has momentum k = (p1−p′1) = (p2−p′2).This meson doesn’t satisfy the usual energy dispersion relation, because k2 6= m2: the
meson is called a virtual particle and is said to be off-shell (or, sometimes, off mass-
shell). Heuristically, it can’t live long enough for its energy to be measured to great
accuracy. In contrast, the momentum on the external, nucleon legs all satisfy p2 =M2,
the mass of the nucleon. They are on-shell. One final note: the addition of the two
diagrams above ensures that the particles satisfy Bose statistics.
There are also more complicated diagrams which will contribute to the scattering
process at higher orders. For example, we have the two diagrams shown in Figures
10 and 11, and similar diagrams with p′1 and p′2 exchanged. Using the Feynman rules,
each of these diagrams translates into an integral that we will not attempt to calculate
here. And so we go on, with increasingly complicated diagrams, all appearing at higher
order in the coupling constant g.
– 62 –
Note that for k = p1 − q1 or k = p1 − q2, k2 < 0 and henceinteraction exchange a virtual particle.
Newtonian force: Yukawa potential
Assume that M m and take non-relativistic limit |~p| M andhence |~q| M , we have (from the center of mass frame) that
〈~q|U(r)|~p〉 = i∫d3xU(r)e−i(~p−~q)·r = − iλ2
(~p− ~q)2 +m2,
where λ = g/(2M). This implies that
U(~r) = −λ2∫
d3p
(2π)3
ei~p·~r
~p2 +m2.
= − λ2
4πre−mr .
The interaction requires an interchange of the virtual meson par-ticle.
The Coulom’s law can be also obtained from QED using thesimilar but more complicated procedure. The force is mediatedby a virtual photon.
Summary of QFT
• The fundamental nature of matter is field.
• The quantum states of the field is particle-like, since theirinteractions are all local.
• Particles, quantum excitations of the same field, are neces-sarily identical.
• The bilinear terms in the action describe free fields; thehigher-order terms of the fields in the action describe theirinteractions.
• Newtonian force is an emergent concept in the non-relativisticlimit.
• We do not have explanation of the existence of matters likeelectrons, but gauge fields and their interactions with matteremerge as localization of some continuous global symmetry,that can be abelian or non-abelian.
• The decaying of a particle does not necessary imply internalstructure.
• Fundamental physics obeys a very precise mathematical rule.
Gravity as QFT
The Lagrangian for Einstein’s theory of gravity is given by
L =√−g(R+ Lmat) ,
leading to the Einstein’s equation of motion
Gµν ≡ Rµν −1
2Rgµν = Tmat
µν .
The action is
S =∫d4xL .
This is the same as the usual quantum field theory, but withthe local general coordinate invariant symmetry, rather than theusual global Lorentz symmetry. Here the fundamental fields arethe metric gµν(x), (or the vielbein+connection in the first-orderformulation.)
Gravity as gauge theory
Einstein gravity arises as localization of the global Lorentz trans-formation. For example, we start with
L =1
2(∂φ)2 − V (φ) .
It is invariant under
x′µ → Λµνxν , Lorentz transformation
If we would like it to be invariant also under the local trans-formation with Λµν = Λµν(x), we could arrive at (the simplestdynamical theory)
L =√−g
(R+
1
2(∂φ)2 − V (φ)
).
The minimum coupling between gravity and matter, and hencethe equivalence principle, is the most natural consequence.
In the process of gauging, we are forced to introduce fields gµν(x)that leaves the quantity
ds2 = gµνdxµdxν
invariant under the local Lorentz transformation.
Restore local symmetry with a field
We have seen so far a rather general phenomenon that whena local symmetry is violated, we can restore it by introducinga new field. For examples, The local phase transformation canbecome a symmetry by introducing gauge fields. The measuredefining the Minkowski spacetime is not invariant under localLorentz transformation, but it can be after introducing the metricfunction gµν.
One keeps on hearing news from observational cosmologists thatthe Newton’s constant G may be time-dependent. This wouldviolate the general coordinate invariance of GR. But we can alsoview that this implies that the theory contains a dilatonic scalarφ such that
L =1
16πGNe−φ√−gR+ · · · .
The theory is then general coordinate invariant again, but with
GeffN = GNe
φ ,
that can be time-dependent.
Thus one has to be careful about the context when one debates.
Important difference
In the usual gauging of the global symmetry in QFT, the globalsymmetry is linear, e.g.,
ψ → eiαψ ,
it remains linear after gauging by setting α = α(x).
Although the Lorentz symmetry x′µ → Λµνxν is linear, after gaug-ing with Λµν = Λµν(x), the symmetry becomes (maximally) non-linear.
This important difference makes it that we cannot simply callEinstein gravity as a gauge theory in the usual sense. This isalso why in addition to the connection 1-form Γµν like A andcurvature 2-form Rµν like F in the usual gauge theory, there arealso the metric functions gµν, whose “square root” is the 1-formvielbein ea.
Gravity as QFT
However, the only well-established approach to QFT theory isthe perturbative method. To be concrete, we do not know howto write a close-form expression for
U(t, t0) = T [exp(−i∫ tt0HI(t
′)dt′)] ,
It can only be done perturbative by doing Taylor expansion ofthe exponential function. The perturbative approach only worksat the weak coupling for a small class of theories that are renor-malizable.
The majority of field theories are not renormalizable! Gravity isone of them.
Rule of thumb: The more nonlinear a theory is, it is less possibleto be renormalizable.
One might also consider using lattice approach to calculate thepath integral
∫e−S where S =
∫d4x√gR is is the Euclidean action.
The non-positive definiteness of the Euclidean action makes it afutile attempt.
Quantum gravity
Without the tool of QFT for quantum gravity, all approaches toquantum gravity have difficulty to deal with
• How to flow from UV quantum gravity to IR classical gravity?
• What are the matter contents at the UV?
• How does the UV theory involving quantum gravity + matterflow to the IR, to give rise to the standard model?
String theory can answer the first two question, but cannot givea precise statement at the moment to the third.
A precise answer now to the last question would require that notmuch happens between 103Gev to 1019Gev, which is rather anoptimistic hope.
That the 750Gev bump is actually just a blip in LHC is not goodfor particle physicists, but it is probably better for understandingthe fundamental physics.
Higher-derivative QFT
Why do we restrict ourselves to consider theories with two deriva-tives? Why not higher derivatives? For example,
L =1
2(∂φ)2 +
1
2(φ)2 .
But this equation does not make sense since the dimensions ofthe two terms are not consistent. We need consider instead
L =1
2(∂φ)2 +
1
2α(φ)2 .
The dimension of the coupling constant α is [α] = L2. In QFT,there is no such a fundamental length scale, it is thus very unnat-ural to introduce higher-derivative theories in QFT, unless one iswilling to introduce a new fundamental constant.
The story is different when gravity is involve. The length scale,namely `2p = GN, already exists in the two derivative theory. It isthus unnatural not to consider higher-derivative theories.
All branches of physics satisfy the democracy rule: Under thesymmetry and given physical constants, all terms could happenwill happen.
New approach?
We come back to our earlier question
• What: What are the fundamental constituents of matter?
Constituents considered so far are all particles! To be precise,we have consider QFT whose excitations give rise to particlespectrum.
Why particle, the dimensionless object? What about strings?
Strings
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Particle action in curved spacetime
Consider general spacetime ds2 = gµνdxµdxν. A particle moves ingeodesics classically, has the action
S = −m∫ds = −m
∫dτ√−gµνxµxν , gµν = gµν(x) ,
The square root in the action is problematic in quantum theory;we can instead use a classical equivalent action by introduce anauxiliary field e = e(x):
S0 = −m∫ √−gµνxµxν ⇐===⇒
classicallyS0 =
1
2
∫dτ(e−1gµνx
µxν −m2e) ,
Note that e is auxiliary because its equation of motion is algebraic.
String action
Consider a piece of “test” string moving in D-dimensional s-pacetime ds2
D = gµνdXµdXν, it spans a two dimensional sheet,worldsheet. Thus Xµ = Xµ(τ, σ). It is natural to generalizethe geodesic motion of a particle to the “minimum” area of theworld sheet. How to calculate the area? We need define conceptof distance, or the metric, on the worldsheet as in the particleexample.
Xµ = Xµ(τ, σ) ,
−→ dXµ =∂Xµ
∂τdτ +
∂Xµ
∂σdσ =
∂Xµ
∂ξidξi , ξ0 = τ , ξ1 = σ .
Thus
ds2 = gµνdXµdXν = gµν(X)∂iX
µ∂jXνdξidξj = γijdξ
idξj ,
where
γij = gµν∂iXµ∂jX
ν
is the “induced metric” of the world sheet:
ds22(worldsheet) = γijdξ
idξj .
The area of the worldsheet is then
Area =∫dτdσ
√−det(γij) .
Example: area of round S2 embedding in R3
R3 : ds2 = dX2 + dY 2 + dZ2 ;
and
S2 : X2 + Y 2 + Z2 = 1 :X = sin θ cosφ , 0 ≤ θ ≤ π ,Y = sin θ sinφ , 0 ≤ φ < 2π ,Z = cos θ ,
Let ξ1 = θ and ξ2 = φ. Thus
γij = gµν∂iXµ∂jX
ν = ∂iX∂jX + ∂iY ∂jY + ∂iZ∂jZ ,
γij =
(1 00 sin2 θ
), det(γij) = sin2 θ ,
Area =∫dξ1dξ2
√det(γij) =
∫dθ sin θ
∫dφ = 4π .
Note that here we do not put minus before the determinant sincethe signature is Euclidean.
Nambu-Goto action
We thus postulate that the string action is given by
SNG = −T∫dτdσ
√−det(γij)
= −T∫dτdσ
√−det(gµνdX
µdXν) .
where T is the tension of string, a dimensionful quantity thatdescribes energy per length.
Polyakov action
The “Polyakov” action is given by
SP = −1
2T∫d2σ√−hhij∂iXµ∂jX
νgµν .
Here hij is auxiliary field. The equations of motion of hij arepurely algebraic.
SNG ⇐===⇒classically
SP ,
Due to the absence of the square root, Polyakov action is moreconducive for quantization, as a 2-d QFT.
Gauge fixing
In addition to the worldsheet reparametrization symmetry, thePolyakov action is invariant under the Weyl scaling the worldsheetmetric In addition to the worldsheet reparametrization symme-try, the Lagrangian is invariant under the Weyl scaling of theworldsheet metric
hij → hij = Ω2hij , hij = Ω−1hij ,√−h = Ω2√−h .
This implies that√−hhij∂iX · ∂jX =
√−hhij∂iX · ∂jX, .
This conformal scaling symmetry is uniquely a 2-d property. Themetric hij has three independent components. We can use thereparametrization symmetry to fix two of them
hij =
(h00 h01h01 h11
)→ h11
(−1 00 1
).
The Weyl scaling symmetry implies that we can set h11 = 1.Thus we have
S =1
2T∫d2σ(XµXν −X ′µX ′ν)gµν .
If we further consider Minkowski spacetime, i.e. gµν = ηµν, wehave
S =1
2T∫d2σ(XµXν −X ′µX ′ν)ηµν .
This is the 2-d theory of D numbers of free scalars Xµ, withµ = 0,1, · · · , D − 1. The advantage is clear since the theory haslinear equations of motion that can be solved completely.
Thus string theory now reduces to a 2-dimensional quantum fieldtheory for D numbers of massless scalars. This however raisesan immediate problem. The Lagrangian is
L = −1
2∂iX
µ∂iXνηµν
= −1
2
(− ∂iX0∂iX0 + ∂iX
1∂iX1 + · · ·+ ∂iXD−1∂iXD−1
).
Thus we see that X0 is a ghost-like scalar in the worldsheet, i.e.it is a field that has negative kinetic energy.
Free action and constraints
Recall that for the Polyakov action before we fix the gauge, theequation of motion for hij implies that vanishing of the energy-momentum tensor
Tij ≡ −2
T
1√−h
δS
δhij= ∂iX
µ∂jXνgµν −1
2hijh
k`∂kXµ∂`X
νgµν = 0 .
For our gauge choice hij = ηij, we have
T01 = T10 = X ·X ′ = 0 , T00 = T11 =1
2(X2 +X ′2) = 0 .
Thus the string theory is not simply
S =1
2T∫d2σ(XµXν −X ′µX ′ν)ηµν .
but also with the constraint of vanishing energy-momentum ten-sor.
This constraint provides a tool to remove the ghost modes whencertain critical conditions are satisfied.
Equations of motion
It is convenient to use light-cone coordinates
σ± = τ ± σ , ∂± =1
2(∂τ ± ∂σ) ,
we have
ηαβ −→(η++ η+−η−+ η−−
)= −1
2
(0 11 0
).
The action becomes
S = −T∫d2σ ∂+X
µ∂−Xνηµν .
The corresponding equation of motion is
∂+∂−Xµ = 0 .
The general solution is
Xµ(σ, τ) = XµR(τ − σ) +X
µL(τ + σ) ,
i.e. it is the sum of the right- and left-movers.
Constraints
The vanishing of the energy-momentum tensor becomes
T++ = ∂+Xµ∂+Xµ = 0 , T−− = ∂−Xµ∂−Xµ = 0 ,
T+− = 0 = T−+ is automatic.
Substituting in the solution, we have the constraint on the solu-tion
(∂−XR)2 = 0 = (∂+XL)2 .
Notations:
A2 ≡ A ·A , A ·B ≡ AµBνηµν .
Mode expansion: closed string
We can perform the mode expansion of XµR and Xν
L:
XµR =
1
2xµ +
1
2`2sp
µ(τ − σ) +i
2`s∑
n6=0
1
nαµne
2in(τ−σ) ,
XµL =
1
2xµ +
1
2`2sp
µ(τ + σ) +i
2`s∑
n 6=0
1
nαµne
2in(τ+σ) ,
Note that
∂−XµR = `s
∑
m∈Zαµme
−2im(τ−σ) ,
∂+XµR = `s
∑
m∈Zαµme
−2im(τ+σ) ,
where
αµ0 = α
µ0 =
1
2`sp
µ .
This allows us to put the momentum mode in the same footingas other modes.
Canonical quantization
In terms of the modes, we have
[αµm, ανn] = [αµm, α
νn] = mηµνδm+n,0 , [αµm, α
νn] = 0 ,
Energy-Momentum tensor
T−− = ∂−Xµ∂−Xµ = 2`2s
∞∑
m=−∞Lme
−2mi(τ−σ) ,
T++ = ∂+Xµ∂+Xµ = 2`2s
∞∑
m=−∞Lme
−2mi(τ+σ) ,
where
Lm =1
2
∞∑
n=−∞αm−n · αn , Lm =
1
2
∞∑
n=−∞αm−n · αn .
Virasoro algebra
It turns out that Lm and Lm each satisfies a Virasoro algebra:
[Lm, Ln] = (m− n)Lm+n +c
12m(m2 − 1)δm+n,0 ,
[Lm, Ln] = (m− n)Lm+n +c
12m(m2 − 1)δm+n,0 ,
[Lm, Ln] = 0 ,
where the central charge c is
c = D ,
the number of Xµ, i.e. the spacetime dimensions.
Defining a physical state
Classically, a physical process must have T++ = 0 = T−−. Interms of modes, it implies that Lm = 0 = Lm for all m. Owingto the existence of central charge c, this cannot be imposed atthe quantum level. We can instead impose
Ln|phys〉 = 0 , n ≥ 1 ; (L0 − a)|phys〉 = 0 ,
with the same for Ln.
It turns out that unitary requires that
bosonic string : a = 1 , c = 26 ,
superstring : a =1
2, c = D(1 +
1
2) = 15 , D = 10 .
Massless state
A massless physical state in bosonic string is
|1,1〉 = ξLµξ
Rν ∂+X
µL∂−X
νR e
ipµXµ |0〉 .The physical condition L0 = 1 = L0 implies that
M2 = −pµpµ = 0 .
The physical conditions Ln = 0 = Ln for n > 0 implies that
pµξLµ = 0 = pµξR
µ .
For those familiar with electromagnetism, it is clear that theabove implies gauge symmetry. The tensor product of two mass-less spin-1 field, gives
symmetric & traceless : hµν = ξL(µξ
Rν) , graviton
antisymmetric : Bµν = ξL[µξ
Rν] , tensor
trace : φ = ξLµξ
Rν η
µν , dilaton .
Worldsheet action again
S = −T∫dτdσ
[√−det(hij) + εij∂iX
µ∂jXνBµν
],
hij = ∂iXµ∂jX
νgµν .
Low-energy effective theory
The interacting theory of the bosonic string implies that theeffective theory for the massless fields (gµν, Bµν, φ) at low energyis a gravity theory in D = 26
L = e−2φ(R+ 4(∇φ)2 − 1
12F2
(3)
), F(3) = dB(2) .
String coupling gs = 〈eφ〉.
The low-energy effective theories of superstrings are the corre-sponding D = 10 supergravities.
uuuµµµªªª444nnnØØØ?
String theory captures all the essence of a final theory: It hasonly one parameter, the string tension. Since it is a dimensionfulquantity, it can be chosen to be either 0 or 1.
Everything else is modulus!
String Theory predictions
• Spacetime dimensions D = 10
• Spacetime dynamics: Supergravity, and hence gravity
• Existence of gauge fields (and hence the concept of “force”)
• Existence of chiral fermion (and hence the concept of “mat-ter”)
but more
QFT regards fields as the fundamentals of matter. Particles arethe excited states of the fields. These states are called particlesbecause their interactions are local, and point like.
String theory generalizes the concept of particle to strings. Arematter strings or fields?
If it were about fields, what are the fields whose quantum stateswould give rise to strings? This question leads to string fieldtheory, whose progress is very limited.
So the argument goes on about the fundamental nature of mat-ter: field-like or object-like (particle, string, membrane, etc.)?
Matter and spacetime
Does matter require spacetime? It is hard to imagine how todescribe matter without spacetime in Newtonian physics. In aquantum world, it is not necessary. For example, the simpleharmonic oscillator can be described by H = ω(a†a + 1
2). In thisdescription, there is no need for space, although it is hard not tointroduce time if one consider quantum evolution. (By definition,evolution involves time.) But nothing wrong with a static world.
If matter is assumed as a particle, its configuration space is nec-essarily a manifold that can be mapped to a set of real numbers,which we may call them xµ, The field φa(x) can be viewed as afunction of the configuration space. Thus the action
S =∫d4xL(φa, ∂µφa) ,
integrates over all possible configurations, with a measure.
Thus particle matter in quantum theory necessary predicts s-pacetime. Other way around, it would be also very unnatural tohave any fundamental matter form other than the dimensionlessparticles in a theory where spacetime is a fundamental concept.
Spacetime concept may not make sense for continuous mattersuch as string, since one can ask what hold them together.
String field theory and the emergent spacetime
A fundamental string is itself a manifold, and its configurationspace cannot be described by another manifold (of finite dimen-sions). (Its topology however is easier to describe.) Let us labelit as Cs. (I have no idea what it is, when preparing this lec-ture.) We may introduce a string field φ that is a function ofthe string configurations φ(Cs). Assuming there exists a conceptof continuum in the configuration space and one can define adifferentiation δφ(Cs). Then we may also define a string fieldaction
S =∫d[Cs]L(φ, δφ) .
In this picture, we do not have the concept of spacetime at all,and hence the question as how a string holds itself together doesnot arise. (Recall also that spacetime can be viewed as nothingbut the configuration space for the dimensionless particles.)
When the string can be approximated as dimensionless particles,the string field theory reduces to QFT, and spacetime emerges.
In this way of thinking, there is nothing fundamental about s-pacetime; it is an illusion of low-energy physics.
The worldsheet string theory we introduced earlier is like therelativistic particle mechanics analogue of QFT.
Conclusion
The most fundamental question in Physics remains:
What is the nature of matter?
Extra
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