Warren Siegel's research
Limitations of string theory
(See also "Particles vs. strings"
for a less technical analysis.)
Most of today's research in high-energy theoretical physics is on string theory.
The main motivations for the current investigation are:
- a finite theory of quantum gravity, and
- a unified theory of all interactions, including gravity.
Unfortunately, the strings under scrutiny are almost exclusively those defined
in 10 spacetime dimensions. Recently these strings were discovered to be
membranes in disguise, defined in 11 dimensions, and all equivalent as a result.
This theory has the problems that:
- Membranes are infinite even in perturbation theory (corresponding to
infinities appearing only nonperturbatively in 10D strings, and thus previously
- Getting D=4 out of D=11 requires compactification, and the infinite
in such compactification effectively nullifies any 11D unification (as well as
reproducing the infinite number of parameters that was the essential problem
produced by infinities), and eliminates the constraints for the few
characteristics (such as the dilaton) found from 10D strings.
On the other hand,
relativistic quantum field theory predicts D=4, since renormalizability fails in
The net result is that known string theories have little or no predictive
power. They are thus less a "Theory of Everything" than a "Theory of Nothing".
(The most unified theory of all is a nonrenormalizable one, since it can
have all possible interactions and couplings; in that sense, all
nonrenormalizable theories are "dual" to each other.)
This situation may be likened to the parton model in the late 60's:
Although it seemed to successfully describe many features of hadrons at large
momenta, it was explicitly wrong in all known models until asymptotic freedom
was discovered in the early 70's.
There are two most likely alternative solutions to the present problems of string theory:
- The present string theory (10D) is the wrong one. Some minor modification (big enough so that it would have been missed, but small enough to still leave it tractable) must be made that will make it inherently 4D, & thus predictive. So 10D string theory should be considered as just a "toy model" that has some correct properties, as well as some incorrect ones that need to be identified & fixed.
- The present string theory is just a general framework, in the same way that relativistic quantum field theory, as defined by Poincaré invariance, unitarity, & locality, is too general (& unpredictive) until the condition of renormalizability is added. Then compactified string theory would be expected to include the right theory, as well as many wrong ones, and a fundamental (non-phenomenological) principle needs to be found to differentiate them.
What has string theory done for us?
- If a viable string theory is ever to be found, it may have many properties
in common with the present string theory.
- Many properties of particle theory have been discovered via string theory,
and then reproduced more simply directly in particle theory:
- String theory was first introduced by d'Alembert in 1747! Although
it was nonrelativistic, it was the first appearance of the wave
equation and the d'Alembertian operator, which were the foundations of
electromagnetic and other field theories, special relativity, and
- Two-dimensional supersymmetry (and supergravity) was
discovered in strings, and then reproduced in four-dimensional particle
- The topological expansion (expansion in the inverse of the
number of colors or flavors) was invented in early string theory (dual
models), and later proved important in understanding confinement in
- The Gervais-Neveu gauge, useful in field theory, was found from string theory.
- BRST first-quantization was used to describe string field
theory, and then applied to find free actions for arbitrary field
theories in arbitrary dimensions.
- String theory was used to derive simple expressions for
one-loop amplitudes in particle field theory, after which simpler
methods were found directly from particle field theory.
- In open (super)string theory, (super)gravity is generated at
one loop, as part of the closed (super)string. Recently the same effect
has been found in a particle field theory, with higher derivatives
characteristic of string theory (see below).
- Superstrings require D=10. The same result was found in the above-mentioned higher-derivative theory.
- Bosonic strings require D=26. The same result follows from pure Yang-Mills
theory if we require cancelation of the leading 1-loop divergence.
- Known string theories seem to be all equivalent. Similar
behavior is found is nonrenormalizable particle theories: e.g., a
theory of quarks with 4-fermion interactions can be shown to be
equivalent to QCD under certain conditions. Such features are common to
nonrenormalizable theories, since lack of predictive power allows any
result consistent with the available symmetries.
- In fact, the only property of string theory not found in
particle theory is Dolen-Horn-Schmid duality, which states that summing
poles in one channel gives the same result as summation in another
channel. This is also the only experimentally verified result of string
theory. It is closely related to Regge behavior, which says that
strings are essentially bound states. Strings are expected as bound
states in QCD, but duality in QCD is still too difficult to derive.
Known string theories can be derived as bound states, but from particle
field theories with unphysical, Gaussian propagators.
Most of my research concerns the relationships between string theory and
particle theory, particularly as regards four dimensions and spacetime
In particular, I would like to find a realistic and calculable string theory that predicts four dimensions, avoiding compactification and its resultant loss of predictability.
Quantum field theory predicts D=4;
in particular, QCD exhibits confinement only in D≤4.
But no string theory of nature has yet been found that finds D=4 as a consequence,
rather than enforcing D=4 by hand.
Since string theory, because of its Regge behavior, is fundamentally a theory of bound states, it would be nice to have a theory where all observed particles are described as bound states of fundamental particles, rather than describe hadrons as bound states of quark and gluon strings, which are themselves bound states of...
(Even the non-strongly interacting particles that are massive, which may eventually prove to be all of them, are effectively described as bound states through the Higgs mechanism.)
For both strings and particles the best method for obtaining explicit results
(for more than just low energies) is perturbation theory.
In particular, a practical theory for QCD should allow a simultaneous description of both confinement and asymptotic freedom perturbatively.
There are several different perturbation expansions, or grouping of graphs, that can be applied simultaneously to the usual loop expansion, based on helicity (selfduality), color, gauge covariance, supersymmetry (superspace), and sometimes T-duality.
- Selfduality: Selfdual solutions (e.g., instantons) are important in QCD, and certain 4D string theories are selfdual.
- Selfduality +: Perturbation about selfduality (the helicity expansion) is useful in QCD, and appears in certain other 4D string theories. Both make use of twistors.
- New 4D strings: Strings that describe hadrons are expected to have critical dimension 4, as required by confinement.
- New symmetries: String theory suggests new symmetries of ordinary field theories.
- Short distance modifications to spacetime: Strings suggest modifications to gravity that can replace black holes and the Big Bang with similar nonsingular behavior. Such modifications can be implemented by an appropriate quantization of spacetime, and can account for the appearance of the graviton as a bound state.
- Superspace: Anticommuting coordinates can simplify supersymmetric particle and string theories.
- New string superspaces: The Anti-de Sitter/Conformal Field Theory correspondence can define new superstrings or superparticles in appropriate limits.
relates one formulation of a theory to an equivalent formulation. When
two such formulations have the same form, it is also a symmetry of the
theory. As usual, making a symmetry of a theory manifest can result in
both simplifications and generalizations, as well as clarifying its
Certain strings (N=2 and 4) have critical dimension 4, and thus seem to be those
most relevant to physics. These strings also describe massless particles only,
and are therefore really particle theories. However, they describe only selfdual
theories. Considering the importance of selfdual theories, these string theories may still be
important in understanding how stringy properties can occur in 4D physics.
Many of the interesting solutions to gauge theories are selfdual. The
corresponding selfdual theories are much simpler than the full theories,
and provide a useful way to study some of their properties.
Light-cone gauge for N=2 strings
(with O. Lechtenfeld), '02
- N=2 worldsheet
instantons yield cubic self-dual Yang-Mills
(with O. Lechtenfeld), '97
- Covariant field
theory for self-dual strings
(with N. Berkovits), '97
self-dual sector of QCD amplitudes
(with G. Chalmers), '96
multi-instantons in conformal chiral superspace, '94
formulation of self-dual superstring, '92
- Self-dual N=8 supergravity as closed N=2(4) strings, '92
- The N=2(4) string
is self-dual N=4 Yang-Mills, '92
- The N=4 string
is the same as the N=2 string, '92
General field theory
actions for non-selfdual theories, even those for massive fields, can be
reformulated in terms of selfdual fields, simplifying calculations and leading to
new insights. Any theory can then be treated as a perturbation about
an almost-trivial selfdual theory, a perturbation in helicity.
This expansion is gauge invariant, as are the loop and 1/Ncolor
Certain perturbative amplitudes in gauge theories ("maximal helicity
violating") are much simpler than expected. This phenomenon can be
explained from the fact that these amplitudes also occur in the selfdual
theories. All selfdual amplitudes have been calculated explicitly; they
vanish at more than one loop. All nonvanishing amplitudes occur in the
A new gauge, the "spacecone gauge", is a complex generalization (Wick rotation)
of the lightcone gauge: Instead of using a fixed lightlike vector, it is defined by a
null complex spacelike vector, which is associated in amplitudes with two
massless external momenta. It incorporates all the known simplifications of
external line factors in massless theories by applying them to internal lines as
well: Only physical polarizations appear as fields, each helicity effectively as a
scalar. It leads directly to the standard covariant twistor methods ("spinor
helicity", "Chinese magic", etc.), and explains all the "miraculous cancellations" in
maximally helicity violating amplitudes. Identities that formerly required
supersymmetry for their derivation are now obvious from the Feynman rules.
Twistor superstrings are an explicit realization of the helicity
expansion in tree graphs: Using a worldsheet theory of only one
handedness, and twistors as variables, the expansion in helicity is an
expansion in the number of worldsheet U(1) instantons. This formulation
is manifestly maximally (N=4) superconformal, but describes only
massless states (N=4 super Yang-Mills). It has an extension that allows
loop calculations and can be expressed as the tensionless limit of a
manifestly supersymmetric, QCD-like string,
with N=4 super Yang-Mills also as its partons.
Untwisting the twistor superstring, '04
Simplifying algebra in Feynman graphs,
Part III: Massive vectors
(with G. Chalmers), '01
Global conformal anomaly in N=2 string
(with G. Chalmers), '00
- Simplifying algebra
in Feynman graphs,
Part II: Spinor helicity from the spacecone
(with G. Chalmers), '98
- T-dual formulation
of Yang-Mills theory
(with G. Chalmers), '97
- Simplifying algebra in
Feynman graphs, Part I: Spinors
(with G. Chalmers), '97
Unlike QED, QCD lacks a useful perturbative formulation: So-called "perturbative QCD" is actually a hybrid between true perturbation theory and pure phenomenology, as only the "hard" (large transverse energy) factors of any amplitude are calculated perturbatively, while the "soft" factors are determined only by experiment. (This is not the usual fixing of masses and couplings, but the determination of functions of various energies.) On the other hand Regge theory, and in particular string theory, attempts to calculate such soft processes (and the hadron spectrum) perturbatively; however, the known string theories are not "QCD strings" (although some attempts have been made recently with the random worldsheet -- see below -- and with the AdS/CFT correspondence
Experiment also shows that the range of validity of both Regge theory and perturbative QCD are greater than expected, and have some overlap. This suggests the possibility of models that at lowest perturbative order would accurately describe both hard and soft processes. Such theories would allow complete calculation of any amplitude for the strong interactions in a systematic way, with increasing accuracy at each perturbative order, without requiring poorly defined and difficult-to-calculate "nonperturbative" contributions, like QED for electromagnetism (and to a great extent unified models of electroweak interactions).
Motivated by, and combining the good points of, earlier attempts to incorporate calculability of both asymptotic freedom and confinement within the same formalism, such an approach can be obtained by assuming the usual strings (but applied to 4 dimensions) have multiple values of the tension, all integers (in terms of some unit). Although this model does give realistic predictions, its self-consistency (and the corresponding fixing of its many arbitrary parameters) has yet to be determined.
An alternative to first-quantization of strings on a continuous worldsheet
is to quantize on a random worldsheet lattice. The random lattice approach
replaces the continuous worldsheet with a lattice, whose random nature
reflects the arbitrariness in the worldsheet metric. Each lattice is directly
identified with a Feynman diagram of an underlying particle theory. This
has the benefit of directly relating the string theory to a random-matrix
theory of preon/parton particles that form strings as bound states.
Massless spinors can't be quantized on lattices, so this approach requires
the Green-Schwarz formulation of the superstring.
Relativistic bound state mechanisms include confinement, which is
responsible for the generation of strings from ordinary field theories,
not just in QCD, but also for fundamental strings, and thus gravity.
String theory was originally derived for the purpose of describing
hadrons. With the advent of QCD, these strings are understood as bound
quarks and gluons. The geometry of strings follows from the 1/N
expansion of QCD.
String theory is now generally used to describe quarks and gluons
themselves, as well as gravitons and other fundamental particles, all from
the (near) massless part of the spectrum. The excited states of these
fundamental strings are not observed, in constrast to those of the
hadronic string. The random matrix model following from such strings
describes particles with Gaussian propagators. Consequently, each
Feynman graph is finite. The condition for the critical dimension is found
only after summing graphs, corresponding to integration over the
In contrast, QCD has divergent graphs, which are renormalizable only in
D ≤ 4, and QCD has confinement only in D ≤ 4,
so D = 4
is the critical dimension for QCD, and
(nontrivial) particle field theories in general. (A related fact is
that 4 is the critical dimension for superconformal invariance.) This
suggests that the
four-dimensional nature of physics might be enforced in string theory
for theories whose random-matrix models have the usual
1/p 2 propagators, rather than Gaussians.
Such propagators can be incorporated into string theory by the
introduction of a second worldsheet metric, which acts as Schwinger
parameters for the Feynman diagrams of the random lattice. For the
bosonic string, the underlying theory is (wrong-sign, asymptotically free) φ4.
The four-point vertices of the Feynman diagrams are the light-cones of the
random-lattice worldsheet. Critical dimension 4 follows not only from
renormalizability, but also from T-duality.
- Worldline Green functions for arbitrary Feynman diagrams (with P. Dai), '06
- Random lattice superstrings (with H. Feng), '06
- Quantized tension: Stringy amplitudes with Regge poles and parton behavior (with O. Andreev), '04
Regge trajectories from worldsheet lattice parton field theory
(with T. Biswas and M. Grisaru), '04
invariance in random lattice strings, '96
for QCD-like strings, '96
- Super Yang-Mills theory
as a random matrix model, '95
- Randomizing the
S-matrices are usually written with a specific gauge choice for external line
factors. An alternative is to write the S-matrix in a way that is
gauge covariant with respect to external line factors; since the external lines
represent asymptotic states, this should be the Abelian invariance.
Using this fact limits the form the S-matrix can take, just as Ward-Takahashi
identities (gauge invariance of the effective action) limit the form of the
effective action. This allows a generalization of the background-field gauge
where 3 different gauge choices can be made: 1 for lines inside loops,
1 for lines inside trees, and 1 for external (amputated) lines.
External line factors are still on shell, but satisfy only the (Abelian)
gauge-covariant field equations.
In string theory, this implies that S-matrices should be calculated in a
slightly different way: Vertex operators should still be BRST invariant.
However, the usual vertex operators preserve the gauge condition b0=0.
Relaxing this gauge condition for the unintegrated vertex operators
allows the addition of Nakanishi-Lautrup terms
(b0≠0 but still ghost number 1). BRST invariance then need imply
only the gauge-covariant field equations. The resulting operators then have
momentum-dependent conformal weights.
In particular, for the superstring,
where the 1-loop 4-point S-matrix is 1PI because of vanishing of lower-point
corrections, it is manifestly gauge invariant because it is a term in the
effective action. But the corresponding tree graph is known to have the
same form, up to a factor depending on momentum invariants.
This formalism allows this result to be obtained directly,
by use of gauge-covariant rules.
String theory has a discrete SO(D,D+n) symmetry for D left-handed and D+n
right-handed dimensions called spacetime, or "T-", duality. In the massless
field theory resulting from the low-energy limit of strings, this becomes a
continuous symmetry. This symmetry (or only subgroups) is seen only in
solutions independent of some of the dimensions. The symmetry can be
treated as a spontaneously broken symmetry, restored by the "high-energy
limit" of distances shorter than the coordinate dependence. As with other
spontaneously broken symmetries, it places strong restrictions on actions.
T-duality is also the least understood of the spacetime symmetries.
This field theory can be formulated, independently of string theory, in a
way where this symmetry is manifest, simply by extending spacetime
indices to have 2D+n values. The fields satisfy covariant constraints that
define a new type of geometry, where the Lie derivative is replaced by a
modified derivative implied by the affine Lie algebra of the string theory.
The field theory retains some stringy properties, such as left-right
factorization of Feynman graphs. Particles (fields) related by this
symmetry, such as the graviton (metric), axion (antisymmetric tensor), and
gluon (Yang-Mills) can be treated as part of a single field. The dilaton,
invariant under this symmetry, is required as the integration measure. In
general this symmetry is spontaneously broken, but for states that are
independent of d dimensions, an SO(d,d+n) subgroup is restored. Similar
remarks apply to string theory, by use of the Hamiltonian form of quantum
mechanics. The method has also been generalized to supersymmetry.
- Gauge-covariant S-matrices
for field theory and strings (with H. Feng), '04
vertex operators (with H. Feng), '03
- Curved extended
superspace from Yang-Mills theory a la strings, '95
- Superspace effective
actions for 4D compactifications
of heterotic and Type II superstrings
(with N. Berkovits), '95
duality in low-energy superstrings, '93
- Superspace duality
in low-energy superstrings, '93
formalism for string-inspired axionic gravity, '93
One convenient ultraviolet cutoff is the lattice. Unfortunately, it breaks
Lorentz invariance by specifying preferred axes, and so cannot be considered
realistic, but only a regularization, to be removed at the end of the calculation.
(An exception is random lattices
, describing discretized curved space.)
A related alternative is to choose a compact momentum space: For the regular
spacetime lattice, momentum space is a torus. Snyder proposed
choosing a sphere instead; then
Lorentz invariance is preserved. (Of course translation invariance isn't,
otherwise we wouldn't have a lattice.) Identifying position operators with
momentum translations, we find they don't commute: The eigenstates of any
one spacetime coordinate are discrete, but different ones are not
simultaneously measureable. The generalization of this concept to supersymmetry
is de Sitter (really spherical) superspace, with fermionic coordinates that
don't anticommute. Fermions are found to live at points halfway between the bosons.
This may provide a solution to the long-standing problem of putting fermions
on a lattice.
Bound-state gravity is the only known
viable alternative to string theory as a quantum theory of gravity.
The most interesting feature of string theory, besides (Dolen-Horn-Schmid)
("stretchiness" of the worldsheet), is that the graviton (closed-string sector)
appears as a bound state (of states of the open-string).
The most remarkable feature of this mechanism is that the graviton is
generated as a bound state at one loop. The explanation is that
closed strings (including the graviton) appear as bound states in free
open-string field theory. This mechanism is analogous to bosonization in
theories of 2D fermions (such as the Schwinger model), where
fermion-antifermion pairs generate bound-state scalars in the free theory,
but their effect is most noticeable in one-loop graphs. Since the
description of the graviton arises from kinematics, and not dyanamics,
open string field theory cannot be interpreted as a theory of bound-state
gravity. However, the corresponding random lattice model can, since
the closed string then results from summation of parton diagrams at arbitrary loops.
Although the phenomenon is purely kinematic in open string field theory,
a better understanding might allow generalization to a true dynamical effect.
A simple (Lorentz covariant) higher-derivative generalization of ordinary
field theory for spins ≤1 is sufficient to reproduce the string effect.
The usual conditions on analytic properties (also applied in string theory)
then reproduce the superstring conditions of D=10 and supersymmetric Yang-Mills.
Besides its nonrenormalizability, the main deficiency of general relativity
is the appearance of singularities in any solution where matter is contained
in a sufficiently compact region. These "black holes" would be unavoidable
in situations known to occur in the gravitational collapse of stars;
similarly, the Big Bang would necessarily begin with a singularity.
However, a fundamental property of relativistic bound states, "Regge behavior",
implies that forces carried by bound states weaken at short distance, where
their bound-state nature emerges. Thus, in any theory where the graviton appears
as a bound-state, including string theory, black holes do not form, and
the consequent problems of singularities and information loss are avoided;
in cosmology, the Big Bang starts from a nonsingular minimum,
which is preceded by a contraction that may account for the features of inflation.
- Non-perturbative gravity, Hagedorn bounce and CMB (with T. Biswas, R. Brandenberger, & A. Mazumdar), '06
- Bouncing universes in string-inspired gravity (with T. Biswas & A. Mazumdar), '05
- Snyderspace (with M. Hatsuda), '03
- Stringy gravity at short distances, '03
- Bound-state gravity from higher derivatives
(with K. Lee), '03
gravity in open-string field theory, '93
The superspace formulation of supersymmetric field theories has resulted in
improvements with respect to calculating classical and effective
actions (supergraphs), and understanding renormalization properties
(finiteness, renormalization group, etc.). So far this formulation is
fully understood for simple supersymmetry (N=1), but only partially for
the extended cases (and higher dimensions), and in particular
N=1 superspace methods have proven useful not only for deriving general
properties of supersymmetric theories at both the classical and quantum level,
but also for explicit calculations of actions and Feynman diagrams.
Extended supersymmetric theories are known to have even more useful features,
but the corresponding superspace methods are still rather primitive.
A problem related to all the above is the incorporation of superspace
into the (super)string. Any string needs fermions to be realistic,
and known fermionic strings require supersymmetry.
(Ordinary field theory may also require supersymmetry to avoid
problems with resummation of the perturbation expansion.)
Manifest supersymmetry (superspace) requires fermions that are
spacetime spinors. The covariant quantization of this
formulation is not fully understood. The simplest approach introduces an infinite pyramid
of ghosts, as implied by extra fermionic (ghostly) dimensions.
Related topics include massive gravity, and higher spins in de Sitter space.
- Simpler superstring scattering
(with K. Lee), '06
- Conquest of the ghost pyramid of the superstring
(with K. Lee), '05
- N=2 harmonic
superforms, multiplets and actions
(with T. Biswas), '01
- Component actions
from curved superspace: Normal coordinates and ectoplasm
(with S.J. Gates, Jr., M.T. Grisaru, & M.E.
- A superspace normal coordinate derivation of the density formula
(with M.T. Grisaru & M.E. Knutt-Wehlau), '97
- The big picture
(with N. Berkovits and M.T. Hatsuda), '91
One approach to this problem comes from string theory:
The AdS/CFT correspondence is the relation ("duality")
between superstrings on an AdS5
and 4D N=4 superConformal Field Theories.
Although the strength of this relationship is
conjectural, it may provide new insight into the properties of 4D field theory.
In fact, the only practical use of string theory so far has been the
string-independent ideas it has inspired, but in this case it applies
specifically to 4D conformal invariance.
In the QCD string picture, where open strings are mesons, the usual
(Maldacena) correspondence identifies the Yang-Mills vectors of the
dual CFT with massless
ρ mesons. However, there is another interpretation that identifies them
with the true gluons. This approach applies two additional principles
to the usual AdS/CFT correspondence:
- With the random lattice worldsheet, the Yang-Mills fields appear explicitly,
as the random matrix fields.
- The projective lightcone limit/expansion introduces a new "holography",
where not only is AdS5 compactified to flat 4D space, but
also S5 is compactified to a flat (internal) 4D space,
as well as compactifying the fermionic coordinates, to result in 4D N=4
projective superspace off shell.
(In contrast, 4D N=4 harmonic superspace is only on shell,
due to more stringent boundary conditions.)
On shell one obtains the usual action for the
4D N=4 superparticle, known to describe 4D N=4 super Yang-Mills.
In addition to applications to the superstring, this implies new approaches
to quantization of extended supersymmetric 4D theories.
- A new holographic limit of AdS5⊗S5
(with M. Hatsuda), '02
- Superwaves, '02
Radial dimensional reduction: (Anti) de Sitter theories from flat
(with T. Biswas), '02
A new AdS/CFT correspondence
(with H. Nastase), '00
Superstrings on AdS5⊗S5 supertwistor space
(with R. Roiban), '00