A Simple Explanation of Quantum Theory: Another Mode of Philosophical Existence Quantum Theory as Another Model of Realism

Broadly speaking, contemporary philosophy is divided into structuralism and post-structuralism. Structuralism corresponds to what is called Kū (emptiness) in Buddhism, while post-structuralism corresponds to Chū (the middle), Chūgan (the Middle View), or Chūdō (the Middle Way). Strictly speaking, what remains when you subtract structuralism from post-structuralism is the Buddhist concept of the Middle Way. In a single phrase, it means to view all ways of thinking through meta-cognition—to see them in relative terms. Examples of different ways of thinking include the classical physics or classical mathematical realist view of things, and the Buddhist Middle Way or structuralist view of things. Besides the classical mathematical and physical perspectives, there is another realist way of viewing things: the quantum theoretical perspective. Philosophy, in its narrow sense, consists of ontology (the theory of being) and epistemology (the theory of knowledge). The mode of realism we learn intuitively and before higher education is a Cartesian, or in other words, modernist epistemology and ontology, which is classical in its physics and mathematics. Quantum theory refines classical physics and offers a comprehensive perspective that includes classical physics. You can view it as either realism or structuralism, but since physics itself is fundamentally realist, it might be more conventional to see it as a form of realism. The same can be said for classical physics; while it can be interpreted structurally, the common understanding is that it presupposes the reality of nature, making a realist interpretation more conventional. Here, the purpose is not to explain philosophy but quantum theory. For those who can only see the world through a classical lens, learning about quantum theory can be beneficial. Therefore, I will provide a simple explanation of quantum theory using only words, without mathematical formulas. Before Studying Quantum Theory I apologize to those who are eager to dive straight into quantum mechanics or quantum field theory, but first, I will briefly explain the general state of current physics. Again, I will use words instead of mathematical formulas. Whether you call it a metaphor, inference, analogy, or expedient means, it is crucial to grasp the meaning of a theory, even if it's described by equations—perhaps a broad understanding is even better. It is important to comprehend it not just intellectually, but to have it resonate with you, to feel it "click," whether through natural language, imagery, or narrative, for both yourself and others. This is the stance I will take, in line with the title, "A Simple Explanation of Quantum Theory." However, while I won't present the formulas themselves, I will provide explanations that are like descriptions of the formulas. My approach is to explain how to conceptualize these formulas through images and words, rather than leaving them as mere equations. I believe it's better to understand the meaning of concepts like the Schrödinger equation or the wave function, don't you think? Is Physics Also Reaching Its Limits? Philosophy is a completed discipline. This doesn't mean philosophy is no longer necessary in the world. Rather, the foundational theory of philosophy, its very groundwork, has been finished. With post-structuralism, contemporary philosophy has completed its foundational theory. What remains are applications of philosophy, its history, and its relationship with sociology—the core is finished, and research has shifted to peripheral areas. Therefore, while many universities retain the old department name "Philosophy," what they are actually doing is ethics, which broadly includes philosophy. In the sense that its foundational and core parts are complete, philosophy is a "finished" discipline. But in a slightly different sense, there is another field where the discipline seems to be at a standstill, unable to progress. This is the area that deals with the fundamental parts of physics. Physics, the Champion of Natural Science The foundation of natural science is physics. There are various fields in natural science, such as chemistry, biology, and earth science, offered as high school electives or specialized departments in university science faculties, but physics is the foundation of them all. Within physics, the areas that deal with fundamental aspects, such as particle physics and cosmology, form the core of the discipline. Applied fields like electromagnetism, thermal-statistical mechanics, and fluid dynamics may exist, but the foundational theory of physics asks the ultimate questions: What is electromagnetism? What is heat? What is force? What is a fluid? The Foundation of Physics is Made of Quantum Theory Quantum mechanics is famous, but there are also relativistic quantum mechanics, which incorporates the special theory of relativity, and further developments like quantum field theory. Quantum theory provides a detailed picture of the microscopic world that forms the basis of our universe. It successfully describes the behavior of electrons, protons, neutrons, and other elementary particles that constitute atoms, and it offers a model more fundamental than atomism, known as the Standard Model of particle physics, which explains what elementary particles the world is made of. Furthermore, while these are theoretical systems that do not include gravity or general relativity, numerous unified theories have been proposed, such as string theory, loop quantum gravity, and M-theory (a more generalized and extended version of string theory), which do incorporate general relativity and gravity. There are several stances. Modern physics, excluding the most cutting-edge research, is composed of quantum field theory and the Standard Model of particle physics. The central concept is the "field," and particles are seen as excitations, or "bits," that arise from parts of this field. Elementary particles can be considered parts of a field. There are various fields in the universe, and each gives rise to a specific type of elementary particle. These particles interact with other fields and the particles that arise from them. In this way, all fields and particles are interconnected, whether directly or indirectly. The current Standard Model and quantum field theory have been refined through verification, experimentation, and theory-building. However, because they cannot yet explain everything, such as the handling of gravity and general relativity, various theories are being developed to provide a complete explanation. There are diverse approaches: theories that attempt to explain all fields as arising from a single fundamental field; theories like string theory, which posit that tiny strings and their vibrational modes explain everything; and approaches that use group theory to describe the interactions of particles and fields, attempting to create a unified description by constructing larger, more extended groups, which in the process might predict new fields based on the symmetries of nature. In this article, I will focus on explaining quantum mechanics and quantum field theory within the broader scope of quantum theory. I will also touch upon relativistic quantum mechanics, which combines quantum mechanics with the special theory of relativity. The Dilemma in Physics: The Limits of Verification and Measurement Unlike the statement "Philosophy is a completed discipline," we cannot say "Physics is a completed discipline." However, it is in a state of gridlock, finding it difficult to advance. This is related to the limits of verification and measurement. To artificially observe phenomena in the microscopic world, scientists conduct collision experiments using particle accelerators to see what phenomena occur and whether new elementary particles can be found. However, there is a limit to the energy levels that humans can currently achieve with particle accelerators, meaning we can only observe phenomena within that range. The results from such observations are often well-explained by existing physical theories. Another approach is to observe phenomena occurring in the vast universe. This involves using astronomy to observe black holes or using detectors like Japan's Super-Kamiokande to capture cosmic rays in an attempt to measure new phenomena or particles. Despite various innovations, there are practical limitations due to budgets and other "real-world" adult circumstances. For these reasons, even if we can create theories that comprehensively explain known phenomena and measurement results, further progress is difficult. This is the current situation for the foundational part of physics. Of course, applied physics—the various sciences in university science departments, other fields of physics, and applications in engineering, technology, and industry—is making remarkable progress. But a sense of stagnation hangs over the effort to delve deeper into the foundations of physics itself. Still, the possibility of a breakthrough cannot be denied. A Brief History of Physics Physics is the foundation of natural science. One might even say it is the foundation of all science, but bridging the gap between the humanities and physics is still a long way off. Philosophy is a humanistic science. From the perspective of modern philosophy, quantum mechanics can be seen as another model of ontology and realism, alongside the classical mechanical worldview. What we learn intuitively during our development, or within the scope of primary and secondary education, is this classical mechanical worldview. It is built upon the concepts of time, space, and matter from figures like Galileo, Newton, and Descartes. Einstein's theory of relativity changed the concepts of spacetime, mass, and energy, but it can be seen as an extension of classical mechanics. By the way, the "dynamics" in mechanics can be read as "change." In a word, mechanics can be called the study of change. Physics itself, in a word, can be called the study of the changes of various things. Therefore, both classical mechanics and quantum mechanics can be said to be studies of the changes of things. The difference is that classical mechanics is based on a classical framework, while quantum mechanics is based on a quantum theoretical framework. Thus, if you want to study quantum mechanics, it is useful to know quantum theory as a general theory. The Word 'Quantum' Doesn't Encompass All of Quantum Theory Judging by the term "quantum theory," one might think it's simply the idea of viewing things as "quanta." That's not wrong, but as quantum theory developed, many things were discovered that differ from the conventional wisdom of classical physics. We need to consider these as well. In other words, I think it's fair to rephrase classical theory and quantum theory as modern physics and contemporary physics. Visually speaking, just as calculus (differential and integral calculus) created classical mechanics, classical mechanics tends to view nature in terms of continuity. In contrast, quantum theory has aspects that view nature discretely. For example, a value that is continuous in classical mechanics might have a minimum value in quantum mechanics or can only take on integer or rational multiples of that minimum. However, the word "quantum," as in "quantization," has come to have a much broader range of meanings as quantum theory has developed. This is because phenomena completely different from classical physics have been discovered. For example, in quantum theory, an electron has wave-like properties. In classical theory, an electron is treated as a particle and does not have wave-like properties. For instance, in classical physics, light is a wave and thus has an image of continuous extension. But in quantum mechanics, light is both a wave and a particle, so it is an entity without continuous extension, and its existence is not continuous. This is the classical physics way of thinking: waves and objects (particles) are separate things. Waves can be superimposed. Conversely, you can define a wave as something that can be superimposed. Because they can be superimposed, waves with periodic phases, like sine waves, can create interference patterns. Objects, whether they are point particles or rigid bodies with volume, cannot be superimposed. Therefore, interference should not occur. In quantum theory, electrons do exhibit superposition. That is, they interfere and create interference patterns on a screen. This means that electrons possess both the properties of classical particles and the wave-like property of superposition. The key point is that this applies not only to electrons but to all elementary particles. To explain this dual nature of particles and waves that all particles possess, equations and functions are used that are modified versions of those in classical mechanics. Explaining these equations and functions is the first primary goal in understanding quantum mechanics. The second major goal is to modify the equations and functions for particles to create equations and functions that describe a field itself. And then, to interpret them. At this point, the protagonist of quantum theory shifts from elementary particles to fields. An elementary particle is nothing more than a part of a field that arises, disappears, interacts, or interacts with other fields and particles at various places and times. But for now, it's best to start by explaining the simplest conceivable model: the Schrödinger equation, which describes the behavior of an electron "orbiting" a proton, and its solution, the wave function. Schrödinger's Quantum Mechanics Many people may have encountered quantum mechanics in high school physics or as a general education course in a science-related field. Those in the humanities or in university departments unrelated to physics may have never taken a course on it. However, it is likely touched upon in chemistry, biology, and even earth science. Quantum mechanics can be expressed in various ways. The most famous are the Schrödinger equation, Heisenberg's matrix mechanics, and Feynman's path integral formalism. The most familiar is probably the Schrödinger equation and its solution, the wave function. I will use this for my explanation. The Schrödinger equation and its solution, the wave function, are used to describe the state of an electron bound to a single proton. In other words, they are for observing the various physical quantities of the single electron in a normal hydrogen atom. In classical mechanics, you would set up an equation to find the motion of the Moon around the Earth, just as you would for the Earth around the Sun. This is a differential equation. Classical mechanics was originally an invention, or discovery, of Newton's. Newton himself, in his famous book Philosophiæ Naturalis Principia Mathematica, described it using Euclidean geometry. He avoided using the calculus he had invented. However, calculus, and more broadly, analysis, along with astronomy and classical mechanics, developed at a tremendous pace, and problems in mechanics came to be solved using analysis. By the way, think of the "dynamics" in mechanics as the cause of change, and "mechanics" as the study of change. The field of study that uses analysis to handle mechanics is called analytical mechanics. This gave birth to a discipline that solves mechanical problems using analysis and algebra. The problem of mechanics is reduced to setting up a differential equation and finding its solution, which is a function. Classical mechanics does not contradict quantum mechanics. You can solve problems of classical mechanics using quantum mechanics. In other words, quantum mechanics is an extension and generalization of classical mechanics. However, when describing the motion of microscopic particles like electrons, strange behaviors are observed that do not appear in classical mechanics for celestial bodies like the Moon or Earth. Physics needs to create theories that do not contradict measured physical quantities. Whether it's an electron, a proton, or a photon, anything that exists in the microscopic world has both particle and wave properties. A theory needs to be constructed to explain both of these. Two successful explanations for the behavior of an electron bound to a proton were Schrödinger's wave mechanics and Heisenberg's matrix mechanics. It has since been shown that both are just different ways of expressing the same thing. To explain Schrödinger's method, it takes the approach of solving a certain equation to find its solution, a function, much like in analytical mechanics. This solution is called the wave function and contains information about the electron's various states, such as its position, momentum, and energy. In physics, observable quantities, whether position or momentum, are called physical quantities. In classical mechanics, it is possible to measure a definite value for a physical quantity, excluding errors. You simply solve a differential equation, get a function, and plug in the parameters; theoretically, that's the end of it. There might be variations in experiments, but those are considered unavoidable dispersions or errors due to the limitations of the measurement method, which doesn't perfectly match the theory. This is deterministic because it's represented by a simple equation and its solution. In quantum mechanics, that changes. A particle also has wave properties. The nature of a wave is that it can be superimposed and has an unbounded extension. Particle nature, whether viewed as a point or a rigid body with extension, has a clear boundary, and different particles cannot overlap. In classical mechanics, having both particle and wave properties is a contradiction. A particle is a particle, and a wave is a wave; they are separate things. The same entity cannot have both properties, and each is described by its own method. In quantum mechanics, even when we use the word "particle," it also has wave properties. Therefore, a different concept is needed to represent it, and the equations have a different structure. The things that are observed and measured are called physical quantities. Position and momentum are examples. In quantum mechanics, to obtain the physical quantities of an electron, one solves the Schrödinger equation to get the wave function and then applies something called an operator to it. This mechanism allows us to predict the possibilities of the observed results. This separation of the wave function and the operator is the foundation of quantum mechanics. By applying an operator to the wave function, you can simultaneously obtain information about the possible values a physical quantity can take and the probability of that physical quantity being observed. When a measurement is actually made in an experiment, one of those values is measured. And the frequency with which that value appears depends on its probability—some are more likely, some are less likely. Wave Function (State Vector) and Operators If classical mechanics involves solving an equation, getting a function, and plugging in numbers to get a definite physical quantity, quantum mechanics requires one or two extra steps. This arises because, in quantum theory, all existence has both particle and wave properties, a more complex ontology than the classical standpoint. This brings about a transformation in ontology that goes beyond just particle-wave duality or viewing physical quantities discretely. The structure, or the operating system, if you will, is different between classical theory (including the classical theory of waves) and quantum theory. In quantum mechanics, you obtain a physical quantity by applying an operator to a wave function. Mathematically, that's what happens, but its interpretation is crucial. Often in quantum mechanics, an electron is described as something like a cloud with an existence probability. When it is measured in a verification experiment, only a single value is measured. The cloud is not measured. Since it's a single value, it's perceived as if one measured the physical quantity of a particle. By repeating the measurement many, many times, you start to get an outline of this cloud-like mode of existence. This is a common way of understanding quantum mechanics when first learning it. The important question here is: why does it exist in the form of a cloud or mist? This is related to the superposition principle. You might have heard this term before; it's also related to Schrödinger's cat. A decisive difference between quantum mechanics and classical mechanics is the measurement problem. Whether in classical or quantum mechanics, at the moment of measurement, a specific physical quantity emerges, and its value is determined. In classical mechanics, the underlying assumption is that even when not being measured, the particle exists in a way that follows the equations and functions. In quantum mechanics, this premise is completely different. In quantum mechanics, the thinking is that "except at the single point of measurement, the particle (sometimes called a wave packet to emphasize its wave nature) simultaneously takes on all conceivable, possible states." The possible states a particle can take may be few or infinite. It takes on all these states simultaneously. This "taking on all possible states" is the key point. Although I use the word "particle," the term "wave packet" is sometimes used as a compromise to emphasize its wave nature. All these possible states are superimposed, and because of their wave nature, they interfere, strengthening or canceling each other out like interference fringes. For those who have studied the electron orbital theory of molecules or the mechanism of covalent bonds in organic chemistry, you probably learned that the electron cloud or mist doesn't exist uniformly but has denser and thinner parts. This is described by concepts like s-orbitals, p-orbitals, and sp2 hybrid orbitals. The electron that forms a hydrogen atom, which can be said to be bound to a proton, also doesn't exist as a uniform cloud or mist but exists with varying probabilities (densities) at different positions. This is due to the superposition principle (a principle, which in mathematics would be an axiom, the highest grade of law) and its wave nature (being able to be superimposed and having a phase). The Measurement Problem At the moment of measurement, the superposition temporarily collapses, and a measured value, a physical quantity, is determined. In the case of Schrödinger's cat, the cat exists in all states until it is measured. The moment it is measured, its state converges to a single one. Various states are possible, but it will be one of the conceivable, possible states. When measured, it might be dead or alive, but its life-or-death status is decided upon measurement. This is a strange feeling from a classical perspective. Some might not just feel it's strange but think it's impossible. However, that's what quantum theory is. Einstein's reluctance to accept quantum theory, finding it hard to believe that the moon he wasn't looking at might not exist in the night sky, is a similar sentiment. "All possibilities exist simultaneously, with varying degrees of probability"—this is the way of thinking, or rather the principle and philosophy, of quantum theory. As I will explain in the section on quantum field theory, it is possible to formulate quantum theory in a way that expresses the principle of least action from classical mechanics. This is called Feynman's path integral, and it too is based on the idea that all possible paths are superimposed; many paths cancel out, but the remaining paths ultimately reveal the physical laws. There is a line in The Brothers Karamazov: "If God does not exist, everything is permitted." In quantum mechanics, it goes further than being permitted; the quantum mechanical way of thinking is that "all possible states must be realized simultaneously according to their degree of probability." This brings up the question: "What is a measured value?" Some god or natural law dictates that when not being measured, all possible states coexist, but the moment a measurement is made, they converge to a single point. Because it converges to a single point, classical mechanics, or our natural, intuitive perception developed through growth, perceives the world as a classical world-picture (a Cartesian or Newtonian world). According to quantum theory, a pure substance—a particle or rigid body without wave properties—does not actually exist. The Procedure of Quantum Mechanics In quantum mechanics, measuring a physical quantity means applying an operator corresponding to the desired physical quantity to the wave function. This yields a probability amplitude, from which the probability of a certain state being measured can be determined. If you want to find the most expected measurement value from a measurement, you can also calculate the expectation value. All of these are performed through mathematical procedures. Quantum mechanics represents these using complex numbers and complex functions, and both their absolute values and phases have meaning. The reason they are not simple real numbers is, broadly speaking, because of the superposition principle and the fact that existence is fundamentally also a wave. Relativistic Quantum Mechanics I wrote that quantum theory includes classical mechanics when scaled up, but the quantum mechanics of Schrödinger, which only describes the behavior of an electron in a hydrogen atom, does not represent all of classical mechanics. For example, it does not take special relativity into account. Nor does it consider general relativity. When a particle's velocity approaches the speed of light, not only classical physics but also quantum mechanics must obey the special theory of relativity. For this reason, a scientist named Dirac created relativistic quantum mechanics, which incorporates special relativity. This is an extended version of Schrödinger's quantum mechanics. By finding the set of functions that are solutions to the Dirac equation (called a spinor) and applying operators to it, physical quantities are determined. The solution to the Dirac equation is a four-component spinor, consisting of four functions, and these four components have meaning. They predict the existence of up and down spin, as well as the antiparticle of the electron, the positron. The existence of spin was already anticipated, but in relativistic quantum mechanics, spin is derived naturally. The Standard Model of Particle Physics As research into the microscopic world progressed, various elementary particles were discovered. In the mid-20th century, many elementary particles were found through particle collision experiments using accelerators. Theories were then created to explain them. Sometimes, a theory predicts the existence of a certain elementary particle, which is later confirmed by experiment. In this way, current physics has created a "specimen collection" of elementary particles called the Standard Model of Particle Physics. It's unknown whether these elementary particles are the smallest and most fundamental constituents of everything, and it has been debated that we may not be able to go further with current technology and observation levels. But, as it happens, or perhaps perfectly, the Standard Model is an exceptionally good model. If you add to this not the classical field but the theory of the quantum field, quantum field theory, your study of quantum theory, or what is commonly called quantum mechanics in a broad sense, will reach a milestone. Particles and Fields When one learns classical mechanics and then begins studying quantum theory with the Schrödinger equation, an preconception tends to arise that quantum theory is about describing the changes of particles, such as their momentum or position. This is a very difficult problem and one that often occurs when progressing from elementary to higher education. For example, you don't teach set theory or category theory to elementary school students when teaching mathematics. You start with arithmetic, the four basic operations. Even before that, you teach numbers. Trying to teach set theory or category theory to elementary schoolers would probably be not only ineffective but also harmful. This is likely related to the structure of the brain. The same goes for physics. It is probably a mistake to start teaching physics with quantum theory in middle or high school. The correct way is probably to teach classical physics first. However, this is sometimes like teaching a lie. In Buddhism, there is a convenient word for this: hōben (expedient means). "A lie as an expedient means"—this is primary and secondary education. It's not that higher education always teaches the truth; in fact, for science, the "truth" may be something like Kant's "thing-in-itself" or Plato's "idea," forever unattainable. Nevertheless, from the perspective of the standard curriculum in higher education, it can sometimes seem like elementary and middle schools are teaching falsehoods. Therefore, it might be fine for primary and secondary education to be provided in a form that is familiar to children, but this can become a major obstacle to learning when they study the content of higher education. This is one reason why university-level mathematics and physics feel difficult. Children's education starts with what is intuitively easy to understand, but this "intuitiveness" can be a hindrance in many ways, not just in the educational content. When learning quantum mechanics, the common sense of classical mechanics can be an obstacle. When learning quantum field theory, the common sense of quantum mechanics can be an obstacle. If quantum mechanics describes a specific elementary particle like the electron, quantum field theory focuses on the field from which that electron originates. The electron is something that arises from the electron field and is merely a part of the electron field. A part of the electron field is excited and becomes an electron. Even if it is "excited," it doesn't become something else; the electron is still part of the electron field, and the electron field remains. In fact, elementary particles are created and annihilated. Or they can change into other elementary particles. This is true both experimentally and theoretically. I mentioned electrons and positrons in the section on Dirac; particles and antiparticles can interact and become other elementary particles. What you first learn in quantum mechanics is that the particle created when the orbit of a hydrogen atom transitions is a photon. You probably learned about alpha decay, beta decay, and gamma decay in high school. Alpha decay is a type of nuclear fission where energy is released. Helium is produced, but photons are also created as a by-product. Since photons are also elementary particles, the creation of elementary particles occurs. In beta decay, electrons and electron antineutrinos, or positrons and electron neutrinos, are created. Both are elementary particles. Photons can also be created as a by-product here. In gamma decay, photons are produced. This means that through interactions and other processes, elementary particles can change into, be created as, or be annihilated into other elementary particles. Since physics is the study of matter, spacetime, and thus the world and the universe, it needs theories, concepts, and tools to explain all of these things. In Schrödinger's quantum mechanics, time and space were already givens, and the existence of protons and electrons was also a given. That is, they were premises. For physics to develop, it needs theories and ideas that can explain the premises themselves, not just take them for granted. For this and other reasons, a new theory was created by positing quantum fields—the source of elementary particles. It's a strange thing to say, but while in classical theory energy can be thought of as something vague, in quantum theory, it can sometimes be easier to think of energy = particle. This isn't just about classical mechanics; in special relativity, too, it might be good to see energy = mass, where mass is a collection of particles, which is a collection of energy. Elementary Particles and Particles I have used the words "particle" and "elementary particle." Even when I say "particle," in quantum theory it also has wave characteristics, so if you want to emphasize that, you could use words like "wave packet" or "oscillator." A particle, for example, a molecule or an atom, can be divided into smaller parts. A methane molecule, for instance, is made of hydrogen, carbon, and electrons. An oxygen atom or molecule is made of one or two oxygen atoms and electrons. An atomic nucleus can be like hydrogen, with just one proton, but there are also hydrogen atoms with one proton and one neutron. Other atomic nuclei are usually made of multiple elementary particles called protons and multiple elementary particles called neutrons. In addition, ordinary atomic nuclei are made of gluons, an elementary particle responsible for the "strong interaction" that binds protons and neutrons together in the nucleus. An elementary particle, literally meaning the "element of a particle," is the smallest unit of a particle, a more fundamental entity that forms particles. Both electrons and photons are elementary particles. In ancient Greek atomism, the atom was considered the ultimate reality. However, in the case of elementary particles, "ultimate" does not mean ultimate reality in that sense. Elementary particles can be annihilated, created, and interact to produce other elementary particles, or they can produce other elementary particles and be annihilated themselves. If an elementary particle can change into another elementary particle, it is difficult to call it an ultimate reality in the sense of ancient Greek atomism. An ultimate reality would not change into or be annihilated by something else called an ultimate reality. The Standard Model of Particle Physics and Quantum Field Theory There are steps in the development of quantum theory. It started with a model like an electron orbiting a single proton, as described by the Schrödinger equation. The existence of the proton and the electron is assumed as a premise. Relativistic quantum theory is an extension of Schrödinger's model that considers the special relativistic effects that occur when an electron's speed is close to the speed of light. As a result, it predicted the existence of up and down spin as degrees of freedom and the existence of antimatter particles, the electron and the positron. After that, many things happened, but from around the 1950s, particle accelerators were built, and it was discovered one after another in experiments that colliding particles could create different particles. From this, a theory was created that particles like protons and neutrons are not elementary particles but are made of more fundamental elementary particles called quarks. The Standard Model of Particle Physics is something that has been built up through repeated experimentation and theory. It lists all the elementary particles currently measured in experiments and also predicts the existence of other, yet-to-be-found elementary particles. Furthermore, an idea emerged that elementary particles are not the fundamental source, but rather that they are excitations of a part of something called a "quantum field." For example, when the electromagnetic field (photon field) is excited, a photon is produced. When the electron field is excited, an electron is produced. The idea of a "field" also exists in classical physics; you may have learned about the gravitational field, electric field, magnetic field, and Maxwell's electromagnetic field, which unified the latter two. A quantum field is a new concept of a field in quantum theory that has some differences from a classical field. Whether it's the Schrödinger equation or the Dirac equation, they are still theories based on classical physics, extending it by turning the Hamiltonian into an operator or incorporating special relativity. The development from a classical field to a quantum field may have a certain inevitability, but it is a paradigm shift with a huge impact. It is a shift from quantum mechanics, where particles are the protagonists, to quantum field theory, centered on fields. The mathematical formalism for handling the entire field is surprisingly similar. If quantum mechanics predicts physical quantities by applying an operator to a wave function, quantum field theory can predict physical quantities—that is, the probability of a certain value appearing or its expectation value—by applying an operator to something called a state vector. In the case of quantum field theory, there is a field operator that is the source of all operators, from which creation and annihilation operators can be derived, and furthermore, all other operators can also be derived. The existence of creation and annihilation operators means that the excitation of elementary particles can also be expressed by operators. Thus, the field is central, and the elementary particles that arise from it are the result. Furthermore, it can describe the excitation of multiple elementary particles, and it also makes it easier to explain the interactions between particles. This is called quantum field theory. These two ideas, the "Standard Model of Particle Physics" and "Quantum Field Theory," represent a pinnacle of quantum theory. It is a highly complete theory, and regarding the Standard Model, the Higgs boson, which it predicted, was discovered not long ago, as was reported in the news. A Deeper Look into Quantum Field Theory In quantum field theory, there is a field for each elementary particle. A certain elementary particle and its field can interact with other types of elementary particles and their fields. Of course, there are also cases where they do not interact. However, when viewed as a whole, the elementary particles of the Standard Model and their fields are indirectly connected even if they do not interact directly. There is no field that is isolated and does not interact with other fields, either directly or indirectly. The Standard Model is not just a list of elementary particles; it is a catalog that also includes these direct and indirect interactions. The procedure for formulating quantum field theory is known. This is called canonical quantization. In quantum field theory, first, operators are obtained by "operator-izing" the analytical mechanics of classical mechanics. Next, a rule called the canonical commutation relation is introduced, following the model of Poisson brackets in classical mechanics. This rule is a principle, and from it, the uncertainty principle and the conjugate relationships of physical quantities can also be understood. Furthermore, the commutativity or non-commutativity of conjugate physical quantities leads to conservation laws. For example, they are derived in the form of Noether's theorem, leading to the conservation of energy with respect to time, the conservation of momentum with respect to translational motion, and the conservation of angular momentum with respect to angle. Incidentally, unlike in classical physics, conservation laws are not always absolute. For example, the law of conservation of energy may not hold for short periods of time. This is because time and energy are in a trade-off relationship due to the uncertainty principle. Next, the state vector is determined. Broadly speaking, this completes the mathematical foundation of quantum field theory. You can think of the state vector as something similar to the wave function. Or rather, the wave function is a type of state vector. And again, physical quantities are obtained by applying operators. You can think of applying an operator as the mathematical equivalent of the act of "measurement" in a verification experiment. The result, depending on what physical quantities are chosen as the basis vectors, is expressed as a vector space over probability amplitudes, with the existence probability of the eigenvalues as coefficients. The square of the absolute value of the probability amplitude is the so-called probability (also called probability density), so the coefficients of the basis vectors express the probability (and phase, since it's a wave) of that physical quantity. If the eigenvalues are discrete, this takes the form of a vector as learned in linear algebra; if the eigenvalues are continuous, it takes a form expressed by an integral. In an actual experiment, only a single point may be measured, but mathematically, the probability of that measured value appearing is expressed. From there, the expectation value of the measured value that will result from the measurement can also be calculated. Weaknesses of Quantum Field Theory Quantum field theory is an excellent theory. Regarding the Standard Model of Particle Physics, due to the limits of experiments and observations, there may be even smaller, more fundamental elementary particles than what are currently called elementary particles, but for now, we are managing without them. In other words, it has succeeded in explaining many aspects of the world's existence without contradiction, using only the currently recognized elementary particles. However, it also has weaknesses. For example, quantum field theory uses a classical, or rather, a special relativistic spacetime, and it is a theory built upon that. This is incompatible with general relativity. General relativity expresses the relationship between mass, gravity, and space through the curvature of space. Quantum field theory does not incorporate this curvature. From another perspective, its explanation of gravity is insufficient. Force can be defined as the cause of change. Mechanics is the study of change. In quantum theory, it might be better to rephrase force as interaction. There are four forces: the electromagnetic force, the strong interaction, the weak interaction, and gravity. The first three forces can be incorporated into the Standard Model of Particle Physics as photons, gluons, and bosons, respectively, but the graviton cannot be successfully incorporated. From this, a unified theory, or a grand unified theory, that unifies everything has been considered, and several have been proposed. For example, just as Maxwell unified the electric and magnetic fields into the electromagnetic field, there is the idea that there might be a field that encompasses all fields. Another idea is string theory, or superstring theory, which attempts to understand all elementary particles and fields in a unified way through the vibrational modes of very small strings. There is also an approach like an extension of gauge theory, which aims to encompass everything with a larger group that includes the Standard Model, as the transformation of particles due to interaction can be described by group theory, which is the study of symmetry. While these can explain various things as grand unified theories, the problem is that verification is difficult. Even with accelerators, there are realistic limits to the scale of the facility and the energy that can be produced, so it seems various efforts are being made in the direction of searching for the key to verification through cosmic observations and other means. Quantum Theory and Philosophical Ontology and Epistemology As I wrote at the beginning, quantum theory, from a modern philosophical perspective, can be understood using either realism or structuralism. This is no different for classical mechanics or classical mathematics. It's just that older ways of thinking were basically only realist, with the exception of the Mahayana Buddhist theory of emptiness (Kū). Quantum theory is a new way of thinking in physics, but it is a branch of natural science. Since natural science itself has a tendency to unconsciously presuppose realism, it's safe to say that quantum theory is also treated as a type of realism. However, even if we call them both realism, classical physics and quantum theoretical physics have quite different flavors. Classical mechanics isn't about getting a deterministically fixed answer from a system of equations, even without going as far as analytical mechanics. The mathematical formulas are a bit complex, or rather, have a strange form, but each part has meaning: the wave function or state vector that represents the state, and the operator that acts on it. A quantum theoretical state, before measurement, assumes all possible states. It might be a bit different, but one could express this as a cloud of probability. Through measurement, such as experiments or observations, at that instant only, all possible states converge, collapse, and are determined into one state. Just because it was determined does not mean it was that way before the measurement; this is not possible in quantum mechanics. It is also different from our everyday sense that we cannot know the past. Until it was measured, it was actually taking on all possible states simultaneously, with their respective probability densities and phases. This is the ontology and epistemology that quantum theory possesses, which is different from classical theory. In modern philosophy, or in Buddhism, post-structuralism and the Middle View (Chūgan) are meta-cognition; they are the OS. Structuralism, the scientific way of looking at things, the classical physical way of looking at things (since physics is the foundation of natural science), and the quantum theoretical way of looking at things are applications. The practical use of modern philosophy is to increase the number of applications. In other words, it is to enable oneself to have various ways of seeing and thinking. Or, it is to master various ways of processing information and to be able to analyze a subject with fundamentally different ways of thinking. For that, it is important to increase the number of applications. In that sense, I have tried to explain quantum theory, which is a more fundamental and generalized way of thinking that embraces classical physics, yet is not widely known.