The Equation of Everything

Introduction

Throughout history, the laws of physics have been repeatedly explained and unified by deeper laws. Isaac Newton unified earthly phenomena with heavenly phenomena when he discovered the laws of motion and gravitation. Maxwell unified all the phenomena of electricity, light, and magnetism when he published his well-known set of equations. The paradoxical constant c in Maxwell’s equations was one of the motivations for the development of Einstein’s Special Relativity, which unified space and time. Quantum mechanics was developed soon afterward and served as a unified theory of chemistry. Quantum Field theory later unified quantum mechanics with special relativity, and eventually developed into the Standard Model of Particle Physics. Meanwhile, the General Theory of Relativity developed as a unification of special relativity and gravity. These two theories, the Standard Model and the General Theory of Relativity, explain nearly everything that we observe in the universe. But we still must ask why the universe is described by these two theories rather than some other set of laws.

Much effort has been expended by physicists to unify the laws of physics into a “theory of everything” over the last century. Most attempts to unify the two major theories fall in the realm of quantum gravity, in which attempts are made to quantize the gravitational field. This approach may turn out to be successful, but it would still not really be a theory of everything. The question would still remain, why did the universe choose to obey this quantum gravitational theory instead of another? Even if theorists managed to explain the quantum gravitational theory with yet another deeper theory, it would be just another incremental step toward that holy grail of physics, and there would still be more work to do. As long as we can still ask the question, “Why this law instead of another?” we will not have that holy grail.

In order to truly discover the theory of everything, we have to come in through the back door, starting with the deepest questions, so that we can develop a theory that needs no explanation. We need to ask a question so deep, that when it is answered, there are no more questions to ask. The fundamental question of existence, “Why is there something rather than nothing?” seems to be the deepest question of all, and I always believed that answering it would resolve many other questions in physics. In a previous article I presented the answer to that question, without assumptions. In this article, I will derive the foundations of physics from that answer.

Mathematizing Existence

In my article on The Origin of Physical Existence, we derived the quasi-existence or conditional existence of “entities,” which can change and interact with one another upon consideration of additional information. All possibilities are constantly occurring, randomly, and this iterative logical application of possibilities generates the flow of time. The entities with specific existence-preserving properties generate the more stable constituents of existence that build the matter in the universe. In this and following articles we will describe this mathematically and work out how this foundation of existence leads to the laws of physics that we are more familiar with. This article is necessarily mathematical, but there are a few points that anyone should be able to understand.

We will refer to each possible entity as a “state” of existence and denote it by $\lvert \psi\rangle$ , and every reference to existence from now on refers to conditional physical existence or quasi-existence. We will now set up an abstract mathematical representation of these states. This representation allows us to describe sums, differences, and transformations of states. This will allow us to see the relationships between all the different states and how they combine to create the universe as a whole.

We can define maps, or operators, denoted by $\vartheta$ , which map one state to another,

$\lvert \psi'\rangle=\vartheta\lvert \psi\rangle.$

We can define the identity operator as the operator that maps each state to itself. We can define a binary sum operator (+) signifying that both the states before and after it exist, and thus form a compound state. Mathematically,

$\lvert \psi_1\rangle+\lvert \psi_2\rangle=\lvert \psi_3\rangle$

where all three $\psi$ are states. This operation is both associative and commutative. We then define subtraction (-) via

$\lvert \psi_3\rangle-\lvert \psi_2\rangle=\lvert \psi_1\rangle$

where $\psi_1$ , $\psi_2$ , and $\psi_3$ satisfy the addition rule of the previous equation.

We will represent the states in such a way that the operators follow the distributive rule,

$\vartheta(\lvert \psi_1\rangle+\lvert \psi_2\rangle) = \vartheta\lvert \psi_1\rangle+\vartheta\lvert \psi_2\rangle.$

Nothing

We can define a state $\lvert \varnothing\rangle$ that represents nothing, or the absence of all states. The equation something+nothing=something can therewith be rewritten symbolically as

$\lvert \psi\rangle+\lvert \varnothing\rangle=\lvert \psi\rangle.$

Suppose an operator $\vartheta$ maps the nothing state $\lvert \varnothing\rangle$ to another state $\lvert \psi\rangle$ . Then

$\lvert \psi\rangle=\vartheta\lvert \varnothing\rangle=\vartheta(\lvert \varnothing\rangle+\lvert \varnothing\rangle)=\vartheta\lvert \varnothing\rangle+\vartheta\lvert \varnothing\rangle=\lvert \psi\rangle+\lvert \psi\rangle$

which is only possible if $\lvert \psi\rangle=\lvert \varnothing\rangle,$ thereby demonstrating that every operator maps the nothing state to itself. It is therefore impossible for something to come from nothing. From now on, we will denote nothing simply as 0, since it is not really a state at all, but rather the absence of any state.

Vectors

We can define scalar multiplication on the states such that these states satisfy properties of a vector space. We can define an inner product of states $\langle~,~\rangle$ such that $\langle\psi_1\lvert \psi_2\rangle=N$ where N is scalar. The inner product we have defined also introduces the idea of “dual” states such as $\langle\psi\lvert$ . If we want the product to be a real scalar (corresponding to probability or number of states), then the dual vector is simply the Hermitian conjugate of the original vector. This leads to the following identity:

$\langle \psi_1\lvert \psi_2\rangle = \langle\psi_2\lvert \psi_1\rangle^*.$

For any state $\lvert \psi\rangle$ , it is possible to define a “normalized” state $\lvert \phi\rangle=\frac{\lvert \psi\rangle}{\sqrt{\langle\psi\lvert \psi\rangle}}$ such that $\langle\phi\lvert \phi\rangle=1$ . Every possible state is therefore a scalar multiple of a normalized state. For normalized states $\lvert \phi_1\rangle$ and $\lvert \phi_2\rangle$ , it can be shown that the product $\langle\phi_1\lvert \phi_2\rangle=1$ if and only if $\lvert \phi_1\rangle=\lvert \phi_2\rangle$ . Consequently, this product serves the purpose of telling us whether two normalized states are the same.

Any state can be written as an operator acting on an arbitrarily chosen normalized state which we call $\lvert \alpha\rangle$ :

$\lvert \psi\rangle=\vartheta_{\alpha\rightarrow\psi}\lvert \alpha\rangle,$

and in the dual representation:

$\langle\psi\lvert =\langle\alpha\lvert \vartheta_{\alpha\rightarrow\psi}^\dagger,$

where $\vartheta_{\alpha\rightarrow\psi}^\dagger$ is the Hermitian conjugate of $\vartheta_{\alpha\rightarrow\psi}$ . Normalized states satisfy

$1=\langle\phi\lvert \phi\rangle=\langle\alpha\lvert \vartheta^\dagger\vartheta\lvert \alpha\rangle,$

$\Rightarrow\vartheta^\dagger\vartheta=1.$

Therefore, all the operators available to serve as maps between normalized states are unitary. It follows that we can write any one of these operators as $\vartheta=e^{i\theta}$ where $\theta$ is a Hermitian operator.

The Universe

Now let’s consider the physical universe, which we call $\lvert \Omega\rangle$ . As determined in The Origin of Physical Existence, we define the universe to be the sum of all possible states,

$\lvert \Omega\rangle=\sum_\psi \lvert \psi\rangle.$

The sum $\sum$ is defined according to the addition rule we defined previously, and signifies that we are allowing all $\psi$ to exist. Every state $\lvert \psi\rangle$ is a scalar multiple of a normalized state, so the equation can be rewritten as a sum over all multiples of all the normalized states.

$\lvert \Omega\rangle=\sum_\phi \sum_N N\lvert \phi\rangle = \big(\sum_N N\big)\sum_\phi\lvert \phi\rangle.$

The sum over scalar multiples is just a constant (which happens to be infinite). We can define a renormalized state $\lvert \Omega_{ren}\rangle$ which does not include that constant, as follows:

$\lvert \Omega_{ren}\rangle=\sum_\phi \lvert \phi\rangle.$

Every normalized state $\lvert \phi\rangle$ can be related to the arbitrarily chosen normalized state $\lvert \alpha\rangle$ by a unitary operator $e^{i\theta}$ . So the sum over normalized states becomes a sum over unitary operators acting on one primordial state $\lvert \alpha\rangle$ . The state $\lvert \Omega_{ren}\rangle$ can also be written as a scalar multiple $\mathcal{N}$ of a single unitary operator $e^{iS_\Omega}$ acting on $\lvert \alpha\rangle$ , so we now have two equivalent expressions:

$\lvert \Omega_{ren}\rangle= \bigg(\sum_j e^{i\theta_j}\bigg)\lvert \alpha\rangle = \Big(\mathcal{N}e^{iS_\Omega}\Big)\lvert \alpha\rangle.\label{alphaOmegaEquationOrig}$

This “alpha-omega” equation is the foundational equation of a new theory of physics. It encodes all of existence, with $\alpha$ representing the beginning of the universe, and $\Omega$ representing the end. Every possible interaction is described by the action of the operators on the $\alpha$ state. The equation represents the universe in three ways. On the left, we simply denote it with the symbol $\Omega$ . In the middle, it is represented as all possibilities acting on the primordial state denoted by $\alpha$ . On the right, all the possibilities have been consolidated into one operator, and the form is similar to the formulations of the major theories of physics. Our goal is now to determine the structure of $S_{\Omega}$ , which we will call the “Universal Action.”

The Universal Action

“Action” is a profound principle of physics. It can be thought of classically as a quantity that describes how a physical system has changed over time. If we look at all paths that a system could take through time, we find that the one that extremizes (minimizes or maximizes) the action is the true path that the system takes. Some interpretations of quantum mechanics take this very literally, where it is assumed that a particle really does take every possible path through spacetime, but all of the timelines that fail to extremize the action interfere with one another and destroy each other. It can be shown that only the path that extremizes the action manages to survive. This is called the Path-integral Formulation of Quantum Field Theory, and it serves as a clear example from contemporary physics of the natural selection of physical laws.

The action is usually denoted by S and is often expressed as the time integral of the Lagrangian L. The Lagrangian is therefore a quantity that describes the state of a physical system at any point in time. Subdividing further, the Lagrangian density $\mathcal{L}$ is a quantity that describes the state of a physical system at any point in space and time. It is often said that the Lagrangian density is what defines a theory of physics. The Standard Model and General Relativity each have their own specific Lagrangian density from which every aspect of the theory can be derived.

Therefore, the path from existential philosophy to physics is through the action $S_{\Omega}$ . If we can quantify the action and convert it to a Lagrangian density defined over space and time, then we will be very close to connecting with the known laws of physics.

Let us now derive the relationship between $S_\Omega$ , $\mathcal{N}$ , and all the operators $\theta_j$ on the left-hand side of Eq.~(\ref{alphaOmegaEquationOrig}). We will first solve a simpler version of the equation, in which we assume a finite number N of operators $\theta_j$ . We write the finite equation with the dependencies written explicitly as follows,

$\sum_{j=1}^N e^{i\theta_j} = \mathcal{N}(\theta_1,...,\theta_N)e^{iS_\Omega(\theta_1,...,\theta_N)}.$

Opposites and Randomness

Notice that if we reverse the sign of every operator $\theta_j$ on the left-hand side, and then take the Hermitian conjugate of the equation, we end up with the same expression because all the operators are Hermitian. The same should hold true on the right-hand side. Since $\mathcal{N}$ and $S_\Omega$ are both Hermitian, it follows that

$\mathcal{N}(-\theta_1,...,-\theta_N)=\mathcal{N}(\theta_1,...,\theta_N)$

and

$S_\Omega(-\theta_1,...,-\theta_N)=-S_\Omega(\theta_1,...,\theta_N).$

At this point, if our set of operators $\{\theta_1,\dots\theta_N\}$ includes a $-\theta_i$ for every $\theta_i$ , or in other words, if every possibility has an opposite, then we immediately get $S_\Omega=0$ . The action is just plain zero and doesn’t depend on anything, no matter what. This would indicate that nothing happens in this universe. It is what it is and that’s that. There is no time and there is no space because there are no possible changes or interactions. It has no form or characteristics beyond the fact that it exists. It is as close to nothing as physical existence can get.

In the special case of a finite number of operators (or things that can happen), it is possible that some operators will not have an opposite, so we can get around this problem. This is the simpler version where we have decided to start our analysis, but we must be aware that in the full theory we need to include every possibility. However, the problem of opposites is resolved by the random sector of existence described in The Origin of Physical Existence, since some entities can randomly exist or act while their opposites randomly do not. This means that, in the end, only the random operators can have any influence on the state of the universe, because all the non-random deterministic operators are canceled out.

The Equation of Everything

Continuing our analysis, the parities (oddness and evenness) of the functions $\mathcal{N}$ and $S_\Omega$ reveal that $\mathcal{N}$ can only include even powers of the operators, while $S_\Omega$ can only include odd powers. The next step is to make use of the Euler equation $e^{i\theta} = \cos\theta + i\sin\theta$ . Using this, the equation becomes

$\sum_{j=1}^N \cos(\theta_j) + \sum_{j=1}^N i\sin(\theta_j) = \mathcal{N}\cos(S_\Omega) + i\mathcal{N}\sin(S_\Omega)$

The real and imaginary parts (or more correctly, the Hermitian and anti-Hermitian parts) of this equation can be evaluated separately, as follows:

$\sum_{j=1}^N \cos(\theta_j) = \mathcal{N}\cos(S_\Omega),$

$\sum_{j=1}^N \sin(\theta_j) = \mathcal{N}\sin(S_\Omega).$

We now apply the series definitions of the trigonometric functions and define $\sigma_k=\theta_1^k+...+\theta_N^k$ such that the equations become

$\sum_{n=0}^\infty \frac{(-1)^n \sigma_{2n}}{(2n)!} = \mathcal{N}\sum_{n=0}^\infty \frac{(-1)^n S_\Omega^{2n}}{(2n)!},$

$\sum_{n=0}^\infty \frac{(-1)^n \sigma_{2n+1}}{(2n+1)!} = \mathcal{N}\sum_{n=0}^\infty \frac{(-1)^n S_\Omega^{2n+1}}{(2n+1)!}.$

Since $\mathcal{N}$ is even with respect to $\theta_j$ , we can write it as

$\mathcal{N} = \mathcal{N}^{(0)} + \mathcal{N}^{(2)} + \mathcal{N}^{(4)} +\cdots$

where the superscripts denote the order or power of the operators $\theta_j$ in each term. Similarly, we can write $S_\Omega$ as

$S_\Omega = S^{(1)} + S^{(3)} + S^{(5)} + \cdots$

Expanding $S_\Omega$ and $\mathcal{N}$ in this manner and looking at just the 0th order terms in the cosine equation, we find

$\mathcal{N}^{(0)} = \sigma_{0}$

where $\sigma_{0}=\sum_{j=1}^N 1 = N$ . We then examine the 1st order terms in the sine equation, and then the 2nd order terms in the cosine equation, and so on. Proceeding in this manner, we find the first few terms of $\mathcal{N}$ and $S_\Omega$ are

$\mathcal{N}^{(0)} = N,$

$\mathcal{N}^{(2)} = -\frac{\sigma_2}{2}+\frac{\sigma_1^2}{2N},$

$\mathcal{N}^{(4)} = \frac{\sigma_4}{24}-\frac{\sigma_1\sigma_3+\sigma_3\sigma_1}{12N}+\frac{\sigma_1\sigma_2\sigma_3}{4N^2}-\frac{\sigma_1^4}{8N^3},$

and

$S_\Omega^{(1)} = \frac{\sigma_1}{N},$

$S_\Omega^{(3)} = -\frac{\sigma_3}{6N}+\frac{\sigma_2\sigma_1}{2N^2}-\frac{\sigma_1^3}{3N^3},$

$S_\Omega^{(5)} = \frac{\sigma_5}{120N}-\frac{\sigma_4\sigma_1}{24N^2}-\frac{\sigma_2\sigma_3}{12N^2}+\frac{\sigma_1^2\sigma_3+\sigma_1\sigma_3\sigma_1+\sigma_3\sigma_1^2}{18N^3}$

$+\frac{\sigma_2^2\sigma_1}{4N^3}-\frac{\sigma_1^2\sigma_2\sigma_1+\sigma_1\sigma_2\sigma_1^2+\sigma_2\sigma_1^3}{6N^4}+\frac{\sigma_1^5}{5N^5}.$

If we now allow N to be very large, then only terms with the highest order of N in each of the above equations will be of any significance, so that we find

$\mathcal{N} = N+(\textrm{terms of order }N^0) ~\rightarrow~ N$

$S_\Omega = \sum_{n=0}^\infty\frac{(-1)^n\sigma_{2n+1}}{N(2n+1)!}+(\textrm{terms of order }N^{-2}) ~\rightarrow~ \frac{1}{N}\sum_{j=1}^N\sin(\theta_j)$

Now we find that the complex sum becomes

$\sum_{j=1}^N e^{i\theta_j} = Ne^{i\frac{1}{N}\sum_{j=1}^N\sin(\theta_j)}.$

We can divide both sides by N and note that the factor $1/N$ is just the portion of operator space $\Delta[\theta]$ corresponding to each operator. If we now allow N to become infinite, then the sum becomes an integral and the $\Delta[\theta]$ becomes an infinitesimal $\partial[\theta]$ . We will also insert a factor $r_\theta$ for each $\theta$ which is randomly either 0 or 1, since only the random sector of existence can remain from now on. The correct equation is then:

$\int e^{i\theta}\partial[\theta] = e^{i\int r_{\theta}\sin(\theta)\partial[\theta]}.$

which means the “alpha-omega” equation can be more properly written as

$\lvert \omega\rangle=\int e^{i\theta}\partial[\theta]\lvert \alpha\rangle = e^{i\int r_{\theta}\sin(\theta)\partial[\theta]}\lvert \alpha\rangle$

where $\lvert \omega\rangle$ is now the twice renormalized version of $\lvert \Omega\rangle$ . To recover $\lvert \Omega\rangle$ , we need to multiply by an infinite constant. This is equivalent to summing up the same operators an infinite number of times, so that the right-hand side of the equation obtains the infinite constant in its exponent. Thus, $\lvert \Omega\rangle$ can be written as

$\lvert \Omega\rangle = \Big(e^{i\int r_{\theta}\sin(\theta)\partial[\theta]}\Big)^\infty\lvert \alpha\rangle.$

This is what I like to call the “Alpha-Omega Equation,” or the “Equation of Everything.” The beginning is $\lvert \alpha\rangle$ , and the end is $\lvert \Omega\rangle$ . In words, everything that can possibly happen to $\lvert \alpha\rangle$ randomly happens or not, and then everything that can possibly happen to those new states randomly happens or not, and so on forever. This is the universe.

Conclusion

We started from the assumption-free philosophy of existence established in previous articles, and by representing the existing entities and their properties mathematically, we derived an equation that describes the entire physical universe. If this theory is correct, it should agree with what we already know about the laws of nature. Conceptually, this theory is in perfect agreement with the principles of quantum physics, but it is not clear at this point how it can reduce to the laws of physics we are more familiar with. Where is the Standard Model of Particle Physics? Where is General Relativity? Where is our 4-dimensional Lorentzian spacetime? We will tackle these questions in future posts.

Truth Reconciled

The Equation of Everything

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