13-Dimensional Models

Theoretical physicists did not choose the numbers 10, 11, 12 or 13 because they liked them. The numbers kept appearing as answers.

In the 1970s, a theory of vibrating strings turned out to make sense only if spacetime had 26 dimensions. A more refined version needed exactly 10. Supergravity worked best in 11. In 1996 two separate lines of work pointed to 12. And in the same year, the physicist Itzhak Bars at the University of Southern California found algebraic structures in string theory that fit most naturally into 13 dimensions — eleven of space and two of time. ^[1,2]

That last proposal is the subject of this article. It is a real, published, peer-reviewed line of research. It is also a niche one: it is not the consensus view of string theorists, it has not been completed into a full theory, and nothing about it has been tested by experiment. To understand why anyone would take it seriously, we first need to understand how physicists end up counting dimensions at all.

When Dimensions Become Constraints

Dimensions as a consistency requirement

In everyday life, the number of dimensions is something we observe: three directions to move in, one direction of time. In some theories, however, the number of dimensions is not an input but a condition for the theory to make sense.

An analogy: a guitar string can only sustain notes whose wavelengths fit its length. Choose the wrong wavelength and the vibration destroys itself. Quantum theories of extended objects behave similarly. If spacetime has the "wrong" number of dimensions, the theory predicts nonsense — for example, probabilities that are negative, or symmetries that hold classically but break down once quantum effects are included. The mathematics selects the dimension numbers for which these failures cancel out.

Why we do not see the extra ones

If a theory needs ten or eleven dimensions but we only see four, the extra dimensions must be hidden. The standard way to hide them is compactification: they are curled up so small that nothing we can do resolves them. The mathematics of this is described in Kaluza–Klein Reduction. Its lesson is that a higher-dimensional theory can look, at low energy, exactly like a four-dimensional one with extra fields.

Why an extra time is harder

Hiding an extra space dimension is routine. Hiding an extra time dimension is a different matter. Time has the opposite sign in the geometry of spacetime, and that sign causes trouble in quantum theory: an extra timelike direction produces states with negative probability, called ghosts, and it threatens the ordered sequence of cause and effect. Curling a time dimension into a small circle does not solve this; it creates loops in time (see Closed Timelike Curves).

Bars's central claim is that an extra time dimension can be made harmless — but only if it comes together with an extra space dimension and a new kind of symmetry that removes the ghosts. Apply that to the eleven dimensions of M-theory, and you arrive at 13.

How the Numbers Emerged

Each number in the sequence was found by following a concrete problem. The history explains why 13 is an extension of the others rather than an arbitrary choice.

26: the bosonic string

The first string theory, developed around 1970 to describe the strong nuclear force, contained only bosons. In 1971 Claud Lovelace noticed that one of its mathematical pathologies disappeared if spacetime had 26 dimensions. ^[3] In 1972 Peter Goddard and Charles Thorn proved that in 26 dimensions the theory contains no ghosts at all — every state has positive probability. ^[4] In 1981 Alexander Polyakov showed where the number comes from: a quantum effect called the conformal anomaly cancels only when $D = 26$. ^[5]

In simplified form, each spacetime dimension contributes one unit to a quantity called the central charge, while the mathematical machinery that removes redundant descriptions of the string contributes $-26$. Consistency requires the total to vanish:

The bosonic string also contains a tachyon — a sign that its vacuum is unstable — so it is not considered a description of our world.

10: the superstring

Adding supersymmetry — a symmetry pairing bosons with fermions — changes the count. Each dimension now contributes $\tfrac{3}{2}$ units, while the redundancies contribute $-15$:

In 1984 Michael Green and John Schwarz showed that in ten dimensions a second, independent quantum inconsistency, the gauge and gravitational anomalies, also cancels, provided the force-carrying symmetry is of a special type such as $SO(32)$. ^[6] This result convinced many physicists that ten-dimensional superstrings deserved serious attention.

11: supergravity and M-theory

In 1978 Werner Nahm showed that supersymmetry cannot be extended beyond eleven dimensions (with one time) without requiring massless particles of spin higher than 2, which are believed to be inconsistent. ^[7] The same year, Cremmer, Julia and Scherk constructed eleven-dimensional supergravity. ^[8] For over a decade it looked like a dead end, because strings lived in ten dimensions.

In 1995 Edward Witten argued that the five known ten-dimensional string theories are different limits of a single eleven-dimensional theory, now called M-theory. ^[9] When the eleventh dimension is a small circle, one obtains a ten-dimensional string theory; as the string interactions get stronger, the circle grows.

12: F-theory and the (10,2) signature

A second ten-dimensional theory, type IIB, did not fit neatly into this eleven-dimensional picture. In 1996 Cumrun Vafa proposed a twelve-dimensional framework, F-theory, in which two extra dimensions form a tiny torus whose shape encodes the strength of type IIB interactions. ^[10] Vafa noted that certain objects in the theory naturally live in a space of signature $(10,2)$ — ten space and two time dimensions — and explicitly left open whether the twelve dimensions were an auxiliary bookkeeping device or something more real.

Independently, Bars observed that the 32 supersymmetry charges of M-theory can be viewed as a single spinor of a twelve-dimensional space with signature $(10,2)$. ^[11]

13: S-theory

Later in 1996 Bars pushed the same analysis one step further, in a paper he called S-theory. ^[1] The ten-dimensional type IIA and type IIB theories have different supersymmetry algebras. Bars showed that both can be written inside a larger structure built from $64\times 64$ matrices that are the natural spinor matrices of a 13-dimensional space with signature $(11,2)$. In his construction, the T-duality that relates the two string theories corresponds to relabelling some of those 13 dimensions. ^[1,12]

In a 2001 review, Bars summarized the reasoning: once dualities are taken into account, the (11,2) signature looked more appealing than (10,2) because it can accommodate both type IIA and type IIB. ^[2]

Why Is It Possible?

Three separate mathematical facts make a 13-dimensional, two-time model possible in principle. None of them shows that it is true.

1. Counting symmetries

The maximal supersymmetry algebra relevant to M-theory has 32 fermionic generators. Their anticommutators produce $32\times 33/2 = 528$ bosonic generators. The same 528 can be packaged in different dimensions:

The fact that the same numbers fit a twelve-dimensional pattern exactly is what first suggested "hidden" dimensions. ^[1,11] In 13 dimensions with signature $(11,2)$, the natural spinor has 64 components, and the smallest supersymmetry algebra containing the 13-dimensional rotation group is called $OSp(1|64)$. Bars, Deliduman and Minic showed in 1999 that all the known supersymmetry structures of M-theory — in eleven dimensions, in the ten-dimensional string theories, and in certain curved backgrounds — sit inside $OSp(1|64)$. ^[13]

Status: this is a mathematical observation about symmetry algebras. It suggests a hidden structure; it does not demonstrate that thirteen physical dimensions exist.

2. Hiding dimensions by compactification

Extra space dimensions can be hidden by Kaluza–Klein reduction. This is well understood mathematically and underlies all of string phenomenology. It explains how an eleven- or thirteen-dimensional theory could look four-dimensional, but it does not handle the extra time.

3. Removing the ghosts of a second time

This is the genuinely new ingredient, called two-time physics (2T physics), developed by Bars with Deliduman, Andreev, Kounnas and others from 1997 onward. ^[2,14,15]

The idea starts from phase space. A particle is described by its position $X^M$ and momentum $P^M$. Bars required a gauge symmetry, called $Sp(2,\mathbb{R})$, that treats position and momentum as interchangeable partners. Such a gauge symmetry imposes three constraints on every physical state:

Here the dot means the spacetime inner product with signature $(d,2)$ — $d$ space directions and two time directions:

The constraints do three jobs at once. They remove the ghost states. They reduce the effective number of independent dimensions by two — one space and one time — so that the physics seen in any single description is ordinary one-time physics in $d$ dimensions. And they only have non-trivial solutions for one specific signature:

• with no time direction, the only solution is $X = P = 0$;
• with one time direction, $X$ and $P$ are forced to be parallel light-like vectors, which is physically empty;
• with three or more time directions, too many ghosts remain for the symmetry to remove.

Only exactly two time dimensions work. ^[2] In Bars's words, two times is "an outcome of the gauge symmetry, it is not put in by hand." ^[2]

A striking consequence is that different ways of fixing the gauge — different ways of choosing which combination of directions counts as "time" — produce different-looking ordinary systems. The same two-time model can appear as a free massless particle, a hydrogen atom, or a particle in certain curved spacetimes. Bars calls these different "shadows" of one higher-dimensional system. ^[14,2]

Status: this mechanism is mathematically consistent for particles and for some field theories. It shows that a second time dimension is not automatically fatal. It does not show that nature uses one.

Putting it together: 11 + 2 = 13

Apply the two-time rule to M-theory: take its $(10,1)$ dimensions and add one space and one time dimension. The result has signature $(11,2)$ — thirteen dimensions. In 1999 Bars, Deliduman and Minic built a toy model of this kind: a point particle moving in 13 dimensions with $OSp(1|64)$ supersymmetry and additional local symmetries that remove the ghosts of the extra time. Fixing the gauge in different ways reproduces eleven-dimensional, ten-dimensional and curved-space versions of M-theory's symmetries. ^[13]

So "13-dimensional" in this context means: eleven dimensions of M-theory, lifted by one extra space and one extra time dimension that are made physically harmless by gauge symmetry.

Related Proposals

Honesty requires noting that 13 is not the only number proposed in this area.

• 14 dimensions, signature (11,3). In 1997 Bars argued that all extended supersymmetry algebras, including the heterotic and type I string cases, arise from a single equation in 14 dimensions with three time directions. ^[16] Rudychev and Sezgin independently studied superparticle and super-Yang–Mills systems in $(11,3)$ dimensions. ^[17,18]
• Other signatures. In 1998 Chris Hull found versions of M-theory in signatures $(9,2)$ and $(6,5)$, related to the usual theory by dualities that can change the number of time dimensions. ^[19]
• Four plus two. Bars later proposed that the Standard Model of particle physics itself can be written as a two-time theory in $4+2$ dimensions. ^[20]

A search of the literature for other well-developed physical theories specifically in 13 dimensions did not turn up a comparable programme. Bars's S-theory and 2T M-theory are the main references for the number.

The Formal Picture in Brief

A useful summary distinguishes three levels of each number:

All of these are statements about the internal consistency of theories. None of them is an observation.

Evidence

Experimental evidence

There is no experimental evidence for thirteen dimensions, for a second time dimension, or indeed for any extra dimension. Collider and short-range gravity experiments constrain extra dimensions without finding any (see Kaluza–Klein Reduction for the current limits). String theory and M-theory, the frameworks the 13-dimensional models extend, are themselves not experimentally confirmed.

Theoretical support

The support for the 13-dimensional picture is internal to theory:

• the supersymmetry algebras of type IIA and IIB fit together inside a 13-dimensional structure; ^[1]
• the two-time gauge mechanism is mathematically consistent and removes ghosts; ^[2,14]
• a toy M-theory model exists in 13 dimensions. ^[13]

This is suggestive, but it is evidence of a possible hidden symmetry, not evidence of physical dimensions.

Reproduction versus prediction

The two-time formulation reproduces known physics — the massless particle, the hydrogen atom, conformal symmetry, and, in Bars's 2006 work, the Standard Model. ^[14,20] Reproducing known results shows that the formalism is consistent with what we already know. It is not the same as predicting something new.

Bars has pointed to one potential difference: in his $4+2$ formulation of the Standard Model, a term responsible for the "strong CP problem" in quantum chromodynamics is absent, which would remove the motivation for a hypothetical particle called the axion. ^[20] If that argument holds, a confirmed discovery of the axion would weaken the case for this formulation — which at least gives the idea a point of contact with experiment. This remains a theoretical claim that has not been widely adopted.

Limitations and Open Problems

It is incomplete. The 13-dimensional M-theory model of 1999 is explicitly a toy model of a single particle. Bars himself wrote in 2001 that supergravity theories in $(10,2)$ or $(11,2)$ dimensions "have been sought for but never constructed satisfactorily." ^[2]

The extra dimensions are not naive dimensions. In S-theory, the twelve-dimensional form of the algebra contains no twelve-dimensional momentum operator, and in both its 12- and 13-dimensional forms there is only one time-translation operator; the extra structure is visible in the symmetry algebra, not as a space you could travel through. ^[1] Bars's own summary is that the extra space and time dimensions exist "but not in the naive sense." ^[2] Popular descriptions of a "13-dimensional universe" often miss this.

Multiple times raise real concerns. Max Tegmark argued in 1997 that a world with more than one genuinely independent time direction would lack the mathematical property (hyperbolicity) that lets observers predict the future from the present. ^[21] Two-time physics avoids this by making the second time a gauge direction, so that only one effective time evolution is physical. The price is that the second time is never directly observable, which also makes it hard to test.

The choice of dimension is not unique. Proposals in 12, 13 and 14 dimensions, with various numbers of time directions, all appear in the literature. They are not necessarily in conflict — they may be different ways of packaging the same hidden symmetry — but this also means that the number 13 is not a sharp prediction.

It rests on an unconfirmed foundation. Every model discussed here extends string theory or M-theory. If those frameworks do not describe nature, the case for a 13-dimensional extension falls with them.

What Thirteen Actually Tells Us

The 13-dimensional models matter less as a claim about the size of the universe than as an example of how physicists search for hidden structure.

The firm result is mathematical: consistency calculations single out 26 dimensions for bosonic strings, 10 for superstrings and 11 for supergravity, while their symmetry algebras contain patterns that fit naturally into 12 and 13 dimensions with two time directions.

Bars's proposal makes the surprising next step coherent. A second time dimension can avoid ghosts when it arrives with an extra space dimension and a gauge symmetry that removes the unphysical states. Our familiar spacetime could then be one "shadow" of a larger structure, with M-theory as its eleven-dimensional face.

Whether that structure belongs to nature is unknown. There is no complete 13-dimensional theory, no confirming experiment, and no consensus that the extra dimensions are more than powerful mathematical bookkeeping.

The honest summary is the one this collection applies to other speculative physics: the mathematics allows it; nature has not yet been asked in a way that could answer.

References

[1] Bars, I. (1997). "S-Theory." Physical Review D, 55, 2373–2381.
https://doi.org/10.1103/PhysRevD.55.2373 · arXiv:hep-th/9607112

[2] Bars, I. (2001). "Survey of Two-Time Physics." Classical and Quantum Gravity, 18, 3113–3130.
https://doi.org/10.1088/0264-9381/18/16/303 · arXiv:hep-th/0008164

[3] Lovelace, C. (1971). "Pomeron Form Factors and Dual Regge Cuts." Physics Letters B, 34, 500–506.
https://doi.org/10.1016/0370-2693(71)90665-4

[4] Goddard, P., & Thorn, C. B. (1972). "Compatibility of the Dual Pomeron with Unitarity and the Absence of Ghosts in the Dual Resonance Model." Physics Letters B, 40, 235–238.
https://doi.org/10.1016/0370-2693(72)90420-0

[5] Polyakov, A. M. (1981). "Quantum Geometry of Bosonic Strings." Physics Letters B, 103, 207–210.
https://doi.org/10.1016/0370-2693(81)90743-7

[6] Green, M. B., & Schwarz, J. H. (1984). "Anomaly Cancellations in Supersymmetric D = 10 Gauge Theory and Superstring Theory." Physics Letters B, 149, 117–122.
https://doi.org/10.1016/0370-2693(84)91565-X

[7] Nahm, W. (1978). "Supersymmetries and Their Representations." Nuclear Physics B, 135, 149–166.
https://doi.org/10.1016/0550-3213(78)90218-3

[8] Cremmer, E., Julia, B., & Scherk, J. (1978). "Supergravity Theory in 11 Dimensions." Physics Letters B, 76, 409–412.
https://doi.org/10.1016/0370-2693(78)90894-8

[9] Witten, E. (1995). "String Theory Dynamics in Various Dimensions." Nuclear Physics B, 443, 85–126.
https://doi.org/10.1016/0550-3213(95)00158-O · arXiv:hep-th/9503124

[10] Vafa, C. (1996). "Evidence for F-Theory." Nuclear Physics B, 469, 403–418.
https://doi.org/10.1016/0550-3213(96)00172-1 · arXiv:hep-th/9602022

[11] Bars, I. (1996). "Supersymmetry, p-Brane Duality and Hidden Space and Time Dimensions." Physical Review D, 54, 5203–5210.
https://doi.org/10.1103/PhysRevD.54.5203 · arXiv:hep-th/9604139

[12] Bars, I. (1996). "Algebraic Structure of S-Theory." arXiv preprint.
https://arxiv.org/abs/hep-th/9608061

[13] Bars, I., Deliduman, C., & Minic, D. (1999). "Lifting M-Theory to Two-Time Physics." Physics Letters B, 457, 275–284.
https://doi.org/10.1016/S0370-2693(99)00582-1 · arXiv:hep-th/9904063

[14] Bars, I., Deliduman, C., & Andreev, O. (1998). "Gauged Duality, Conformal Symmetry, and Spacetime with Two Times." Physical Review D, 58, 066004.
https://doi.org/10.1103/PhysRevD.58.066004 · arXiv:hep-th/9803188

[15] Bars, I., & Kounnas, C. (1997). "Theories with Two Times." Physics Letters B, 402, 25–32.
https://doi.org/10.1016/S0370-2693(97)00452-8 · arXiv:hep-th/9703060

[16] Bars, I. (1997). "A Case for 14 Dimensions." Physics Letters B, 403, 257–264.
https://doi.org/10.1016/S0370-2693(97)00550-9 · arXiv:hep-th/9704054

[17] Rudychev, I., & Sezgin, E. (1997). "Superparticles in D > 11." Physics Letters B, 415, 363–370.
https://doi.org/10.1016/S0370-2693(97)01307-5 · arXiv:hep-th/9704057

[18] Rudychev, I., Sezgin, E., & Sundell, P. (1998). "Supersymmetry in Dimensions Beyond Eleven." Nuclear Physics B — Proceedings Supplements, 68, 285–294.
https://doi.org/10.1016/S0920-5632(98)00162-5 · arXiv:hep-th/9711127

[19] Hull, C. M. (1998). "Duality and the Signature of Space-Time." Journal of High Energy Physics, 1998(11), 017.
https://doi.org/10.1088/1126-6708/1998/11/017 · arXiv:hep-th/9807127

[20] Bars, I. (2006). "The Standard Model of Particles and Forces in the Framework of 2T-Physics." Physical Review D, 74, 085019.
https://doi.org/10.1103/PhysRevD.74.085019 · arXiv:hep-th/0606045

[21] Tegmark, M. (1997). "On the Dimensionality of Spacetime." Classical and Quantum Gravity, 14, L69–L75.
https://doi.org/10.1088/0264-9381/14/4/002 · arXiv:gr-qc/9702052