**Abstract **

Scientists have been trying to understand the origins and structure of the universe for centuries. It is believed that the Grand Unified Theory, (also referred to as the *Theory of Everything*) would be able to explain every physical phenomenon and interaction in the universe. Given our current understanding, scientists have been able to provide laws that apply to either the quantum world or classical physics. However, it only seems logical that a universal set of laws should apply regardless of the scale. There have been numerous failed attempts to form a Grand Unified Theory, and many scientists believe that it is impossible¹. Although the attempts have not been successful so far, we have gained invaluable insight and understanding through the endeavours to find the Grand Unified Theory. There are several theories being researched currently, and three have gained traction in the scientific community which are: the Standard Model, String theory and Loop Quantum theory.

**Background**

The process of combining laws began centuries ago when mechanics and thermodynamics were combined, followed by Faraday and Maxwell, who combined electricity and magnetism. Einstein developed the concept of General Relativity, which redefined our understanding of gravity previously provided by Isaac Newton². After developing General Relativity, Einstein endeavoured to combine it with the electromagnetic force. There are clear similarities in the behaviours of the two forces, however ultimately Einstein was unsuccessful. As Einstein strove to combine the laws regarding these forces, the concept of quantum mechanics had gained traction in the scientific community. The theory explained interactions on a very small scale. Einstein was unsure of the concept and was particularly sceptical of the theory’s dependence on probability. However, quantum mechanics survived rigorous experimental tests and is often coined as one of the most successful and precise theories to be formulated. Once the theory was verified, the process of combining the theory with existing laws commenced.

**What is the significance of finding the Grand Unified Theory?**

Finding the unified theory might be able to further explain phenomena that aren’t currently understood such as: dark matter, supersymmetric partner particles, antimatter asymmetry and magnetic monopoles³.

**The Standard Model**

The Standard Model describes and classifies all the fundamental particles that make up our universe and how they interact. The idea was formed from our knowledge of subatomic particles in the 1970s and these fundamental particles make up everything around us⁴.

Quarks | Leptons | Bosons | Higgs Boson | ||
---|---|---|---|---|---|

Up | Down | Electron | Electron neutrino | Photon | |

Strange | Charm | Tau | Tau Neutrino | Gluon | |

Top | Bottom | Muon | Muon Neutrino | Z boson | |

W boson |

In the Standard Model, the particles have different properties and the manner in which they interact with each other is what we observe as forces. Quarks are always found in doublets (mesons) or triplets (baryons). The most common hadrons are protons and neutrons found in the nuclei of atoms. Leptons are another type of particle; the electron is the most stable lepton and is found orbiting the nuclei of atoms. The bosons represent the fundamental forces (strong, weak, and electromagnetic) and are involved in particle interactions. Particles have different properties: charge, spin and mass. The Higgs boson is the particle that gives particles the property of mass.

The presence of these atoms was proved experimentally, and the fundamental particles in the Standard Model were able to explain *almost all *particle interactions and physical phenomena in the universe. The Standard Model has been tested extensively and has been proved to be approximately correct regarding interactions down to 1/1000^{th} the size of an atom’s nucleus⁵.

However, the Standard Model has its limitations. Along with the fundamental particles that make up matter, a particle should represent each of the four fundamental forces: (the strong force, the weak force, the electromagnetic force and the gravitational force). Only three have been discovered so far: the strong force carried by the gluon, the electromagnetic force carried by the photon and the weak force carried by the W and Z boson. Scientists have predicted the graviton to carry the gravitational force, but it has yet to be discovered. This particle has been particularly difficult to observe as the magnitude of the gravitational force is proportional to mass; where on such a small scale the force is negligible.

General relativity takes a different approach to the gravitational force and suggests that there may not be a ‘particle’ to represent the force of gravity and it is yet to be incorporated into the Standard Model⁶.

**Background: General Relativity**

General Relativity is Einstein’s formulation of gravity which shows that space and time display the gravitational force through curvature. Einstein discovered that space has a structure similar to that of a fabric that is distorted when large objects are present, and this is what we experience as gravity. The larger the mass of the object, the more the space fabric is distorted, hence a stronger gravitational force will be experienced.

*Figure 1: The distortion of space by mass*

In the image above, you can see how the earth has ‘distorted the space fabric’. The larger the mass of the object, the more it distorts the fabric. This also corresponds to the gravitational pull of the object. The more the fabric is distorted the stronger the gravitational pull. This explains why larger masses have stronger gravitational fields (e.g. Earth has a higher mass and consequently a stronger gravitational field strength than that of the Moon).

**String Theory **

String theory is an attempt at a unified theory which suggests that instead of non-zero point particles making up the universe (fundamental particles in the Standard Model), there are one dimensional filaments called strings. String theory is particularly interesting as it is able to combine quantum mechanics and general relativity, two fields of Physics that seemed incompatible⁷. String theory is seemingly more complex than the Standard Model, requiring a combination of complex maths and physics.

String theory postulates that if we were to zoom in on a particle far beyond that which is within our current technological capability, we would find a one -dimensional string. The previous table of particles dictated by the Standard model would be found when the string vibrates at different frequencies, with each resonant frequency representing a different particle. The different oscillatory patterns of the string would determine the different properties of the particle: charge, mass and spin. This theory is particularly exciting as it is the first theory to link all of the properties of a particle to a single idea – a string.

We experience a four dimensional world on a day to day basis (3 spatial dimensions and spacetime). General relativity showed how dimensions could be curved by mass and two scientists, Theodor Kaluza and Oskar Klein, proposed that there was another spatial dimension. If this special dimension was curled up until its radius was small enough then it wouldn’t be noticed⁸.

*Figure 2: Configuration of five dimensions*

In the image above we can see the curled dimensions curled up on the ‘space fabric’. This showed that our universe contained four spatial dimensions and one space-time dimension. The theory went on to suggest that there were other possible dimensions with small radii. The Kaluza – Klein theory was experimentally verified and with the discovery of the Strong and Weak forces the theory was further developed until two further spatial dimensions were found.

*Figure 3: Configuration of seven dimensions*

Once these globe structures were discovered, two more curved dimensions were designed.

*Figure 4: Configuration of nine dimensions*

The further dimensions formed donut shapes on the space fabric

This showed that it was possible that there are numerous dimensions that we are not aware of and that the dimensions could be different curvatures forming different shapes. Through the development of string theory, it was suggested that there are eleven dimensions curled within each other. The way in which these dimensions were configured was extremely important as they influenced the frequency and the oscillations of the strings meaning that the geometry of the dimensions affects the properties of the particles⁹.

The equations formed from string theory limit the shapes that the dimensions can form. The six-dimensional shapes formed are called Calabi-Yau spaces.

*Figure 5: Calabi Yau space*

Calabi Yau Space (Greene, the Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory n.d.)

The Calabi-Yau spaces are found on the three dimensional space fabric.

*Figure 6: Configuration of Calab Yau spaces in universe*

With these developments, Calabi-Yau spaces were linked to the three families of particles found in the Standard Model. The different holes in the Calabi- Yau spaces allow for low frequency oscillatory patterns of the string – which is where the different families of particles would originate (e.g. quarks and leptons). Therefore, we must find a Calabi – Yau space with three holes to correspond to the different families of fundamental particles. However, one of the limitations of string theory is that there are thousands of different Calabi – Yau spaces, each with a different number of holes. Currently, it is impossible to mathematically deduce which of the Calabi – Yau shapes constitute the extra spatial dimensions necessary to support the Standard Model. With our current limited knowledge of the way in which the oscillatory patterns affect the properties of the particle, we are unable to differentiate between the Calabi – Yau spaces that fit the vague criteria. If none of the Calabi-Yau spaces are suitable once the criteria become more specific, the beautiful theoretical framework will fall apart.

String theory is a very different approach to the Standard Model, however it is encouraging to see how the Standard Model, which has experimental evidence, can be incorporated into String theory. Although we may be far from experimentally verifying the String theory, further research could develop further requirements for Calabi-Yau spaces required for the framework and this may bring us closer to finding the Grand Unified Theory.

**Loop Quantum Theory**

Loop Quantum Gravity was developed in 1986 by Abhay Ashtekhar, with its foundations built upon Einstein Field Equations, and the theory was further developed by Carlo Revelli and Lee Smolin in 1986. Loop Quantum Gravity attempts to combine General Relativity and the Standard Model¹º.

Loop Quantum Gravity is made up of quantised loops of gravitational fields, called spin networks. Loop Quantum Gravity has been quantised down to Planck length, and it has been deduced that within regions the size of a Planck length(1.61×10^{-35} m), there are vacuum fluctuations that are so large that, space ‘boils’ and forms a froth, called quantum foam. However, on a larger scale (10^{-12}m), the space appears to be completely smooth, with the roughness of the space only visible at 10^{-30} m. Further research into quantum foam suggests that it possesses an intrinsic energy that could be incorporated into general relativity equations by acting as a new cosmological constant. Quantum foam could also form part of the explanation as to why the expansion of the universe is accelerating¹¹.

*Figure 7: Structure of space according to Quantum Loop Theory*

Although the theory does not suggest that there are more dimensions, the concept is similar to String Theory as they both suggest that space-time becomes more complex as you look on a smaller scale. Cartesian coordinates cannot be used on such a small scale as Loop Quantum Theory uses noncommutative geometry which is the expression of the uncertainty principle. The Loop Quantum Gravity dictates a minimum linear size of space, the Planck length. However, the length is not the fundamental attribute because the theory is based on angular momentum which directly corresponds to an area element, and this area is more fundamental than the length: the smallest non-zero volume is 10^{-99} cm^{3}. This concept is akin to the idea that matter is made up of small atoms – space is made up of small volumes.

This extended space can be simplified by representing the volume as a point or a node, with the area of the enclosed volume represented as a line perpendicular to the face as seen in Figure a and b. Figures c and d show how these volumes can be connected. Many discrete volumes connected together are called spin networks. Different particles are found at different nodes and fields can be created by adding different labels to the graphical representation.

*Figure 8: Graphical representation of particles and fields*

*/*

Time is also perceived differently. Instead of time being a constant flow, time passes in a similar way to the second hand on a clock, but with each tick being approximately the same as a Planck length 10 ^{-43} seconds. Or, more precisely, time in the universe flows by the ticking of numerous ‘clocks’ – every location in the spin network where a quantum \”move\” takes place, a ‘clock’ at that location ticks once. The lines of the spin network become planes, and the nodes become lines which can be seen in the figure below. This result is called spin foam and a slice of a spin foam at a certain time will result in a spin network. By taking multiple shots over time, the progression of the spin network is revealed¹².

*Figure 9: Progression of a spin network*

For Loop Quantum Theory to be viable, General Relativity should also be found through the theory. We are almost certain that General Relativity is an accurate representation of how gravity functions as it has survived rigorous experimental testing. The theory of special relativity states that an object will appear to contract depending on its relative speed to the observer. The contraction of the object would also affect the size of the space time chunks that are described in Loop Quantum Gravity. This would mean that the size of the space-time chunks would depend on the perspective of the observer which contradicts Loop Quantum Gravity’s perception of gravity. Physicist Jorge Pullin believes that making loop quantum gravity compatible with special relativity would lead to interactions similar to those that occur in string theory.

**Conclusion**

The Standard Model remains as the only attempt at a unified theory with some experimental evidence, however scientists are yet to discover the graviton subatomic particle. Some scientists believe that gravity is different to the other fundamental forces and is therefore unlikely to be discovered. Therefore it is only reasonable that we look to the other theories (String theory and Loop Quantum Theory) for the Grand Unified Theory.

String theory and Loop Quantum Theory are similar in that they both attempt to integrate the fourth fundamental force gravity into small quantum situations. The theories have similar mathematical concepts and the same approach to the problem, as they both form a new framework to fit the different particles from the Standard model. This could mean that the theories could be combined to overcome their individual limitations. Many scientists believe that the collaboration of string theorists and those researching Loop Quantum Gravity could lead to further developments in both theories.

Developing an accurate Grand Unified Theory could allow scientists to gain a deeper understanding of matter and forces, revolutionise the foundations of physics and aid scientists in finding solutions to the questions that remain unanswered. The discovery of the unified could also lead to major developments in technology and other branches of science. It could be that we discover that it is impossible to form a grand unified theory or ‘Theory of Everything’. If such a theory does exist, String theory combined with Quantum Loop theory seems to be the most likely theory to be successful unless the graviton particle (in the Standard Model) is discovered.

# Bibliography

1.Mann, Adam. 2019. *What is theory of everything.* August. https://www.space.com/theory-of-everything-definition.html.

2.Jogalekar, Ashutosh. 2013. *Why the search for a unified theory may turn out to be a pipe dream.* May 3. https://blogs.scientificamerican.com/the-curious-wavefunction/why-the-search-for-a-unified-theory-may-turn-out-to-be-a-pipe-dream/.

3. Siegel, E. (2017, July 8). *Ask Ethan: How Close Are We To A Theory Of Everything?* Retrieved from Forbes: https://www.forbes.com/sites/startswithabang/2017/07/08/ask-ethan-how-close-are-we-to-a-theory-of-everything/#35bb5d1e5e9c

4. n.d. *The Standard Model.* https://physics.info/standard/ .

5. Langacker, Paul. 2012. *Grand Unification.* http://www.scholarpedia.org/article/Grand_unification.

6. *What Is The Standard Model of Particle Physics?* (n.d.). Retrieved from Science Alert: https://www.sciencealert.com/the-standard-model

7. Greene, Brian. n.d. T*he Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory.*

8. Berman, David. 2012. *Kaluza, Klein and their story of a fifth dimension.* October 10. https://plus.maths.org/content/kaluza-klein-and-their-story-fifth-dimension.

9. Greene, Brian. n.d. T*he Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory.*

10. Tate, Jean. 2010. *What is Loop Quantum Gravity?* Jan 14. https://www.universetoday.com/50702/loop-quantum-gravity/.

11. Lindley, David. 2019. *“Quantum Foam” Scrubs Away Gigantic Cosmic Energy *Sep 27

12. n.d. *Quantum Foam and Loop Quantum Gravity.* http://universe-review.ca/R01-07-quantumfoam.htm .

Image references

Figure 1: https://asd.gsfc.nasa.gov/blueshift/index.php/2015/11/25/100-years-of-general-relativity

Figure 2: http://mafija.fmf.uni-lj.si/seminar/files/2011_2012/KaluzaKlein_theory.pdf

Figure 3: https://busy.org/@mike11/what-is-the-klein-kaluza-theory-and-it-s-implications

Figure 4: https://publicism.info/science/elegant/9.html

Figure 5:https://www.daviddarling.info/encyclopedia/C/Calabi-Yau_space.html

Figure 6:https://www.researchgate.net/figure/A-representation-of-a-6-dimensional-Calabi-Yau-space-Ref21-p-207_fig4_1969741

Figure 7: http://universe-review.ca/R01-07-quantumfoam.htm

Figure 8: https://universe-review.ca/R01-07-quantumfoam.htm

Figure 9: https://universe-review.ca/R01-07-quantumfoam.htm