Leonard Susskind gives the first lecture of a three-quarter sequence of courses that will explore the new revolutions in particle physics. In this lecture he explores light, particles and quantum field theory.

Summary: 

• (00:01) The central question of particle physics is whether nature is fundamentally discrete (composed of indivisible particles) or continuous (made of uniform fields). Quantum mechanics ultimately showed that reality is a complex synthesis of both concepts.

• (05:48) The first major evidence for the discrete nature of matter came from chemistry. John Dalton discovered that chemical elements had masses that were integer multiples of hydrogen's mass, suggesting they were all built from common, fundamental units.

• (13:21) The discovery of radioactivity unveiled that radiation is emitted in three distinct forms (alpha, beta, and gamma) and was the first experiment to produce a "particle beam," hinting at the subatomic world.

• (27:23) For a long time, light was considered the definitive example of a continuous wave, a fact demonstrated by interference patterns in experiments like the double-slit setup, where light waves can either reinforce or cancel each other out.

• (41:40) This view was upended by the discovery that light also consists of discrete, indivisible packets of energy called photons. At very low intensities, light arrives as individual "blips" that collectively build up to form the classic wave pattern.

• (47:46) This perplexing wave-particle duality is not unique to light. Electrons, traditionally thought of as particles, also exhibit wave-like behavior by creating interference patterns, a principle that applies to all matter.

• (51:23) The energy of a photon is directly proportional to its frequency (E = ħω). Consequently, short-wavelength light (like X-rays and gamma rays) is composed of highly energetic photons.

• (55:51) Einstein's equation, E=mc², reveals the profound connection between energy and mass, establishing that they are interchangeable. The mass of a particle is simply a measure of its energy when it is at rest.

• (1:44:06) To see smaller and smaller objects, one must use probes with shorter wavelengths. Because shorter wavelengths correspond to higher momentum and energy, physicists must build progressively larger and more powerful particle accelerators to probe the fundamental structure of the universe.

In this lecture Professor Susskind explores quantum field theory.

Summary: 

• (00:05) The lecture introduces quantum field theory as the fundamental tool for understanding particle physics, highlighting the connection between waves and particles through quantum mechanics.

• (01:38) It reviews the essential mathematical relationship between sines, cosines, and exponential functions, which are used to describe wave-like oscillations. The formula e^(iKX) = cos(KX) + i*sin(KX) is presented as a key concept.

• (10:22) The discussion moves to the core principles of quantum mechanics, introducing Planck's constant (h) as the crucial factor that relates a wave's properties, like frequency, to a particle's properties, like energy.

• (14:08, 20:08) The energy of a single photon is shown to be directly proportional to its frequency (E = hf), and its momentum is inversely proportional to its wavelength (p = h/λ).

• (26:26) The amplitude of a wave is connected to the number of photons it contains. Specifically, the square of the wave's amplitude is proportional to the number of photons.

• (53:33) In a finite, periodic space, such as a circle, a particle's momentum is quantized, meaning it can only exist in discrete integer multiples of a fundamental unit.

• (1:07:33) The quantum harmonic oscillator is introduced as a central concept for understanding quantum fields. Unlike its classical counterpart, its energy is quantized and can only exist in discrete levels.

• (1:25:30) The lecture explains creation (a+) and annihilation (a-) operators, which are mathematical tools used to add or remove quanta of energy, moving the harmonic oscillator between its quantized energy states. These operators are fundamental to describing the creation and destruction of particles.

• (1:48:12) A quantum field is ultimately defined as a collection of harmonic oscillators, where each oscillator corresponds to a different mode of oscillation (frequency).

In this lecture Professor Susskind talks about what a quantum field is and how it is related to particles.

Summary: 

• (00:05) The lecture begins by establishing the harmonic oscillator as the fundamental mathematical tool for understanding quantum fields.

• (00:54) To simplify the mathematics and maintain momentum conservation, the concept of a periodic space is introduced. This is a finite space that loops back on itself, allowing waves to travel without reflection.

• (02:14) On this periodic space, the wave number (k) of a particle becomes quantized, meaning it can only take on discrete integer multiples of a base value. This is directly related to the particle's momentum.

• (10:24) The idea of "occupation numbers" is introduced to describe how many particles, or quanta, exist for each possible quantized momentum state.

• (13:18) Creation (a+) and annihilation (a-) operators are defined. These are mathematical tools that, when applied to a quantum state, increase or decrease the number of particles in a specific momentum state.

• (19:13) A quantum field (ψ) is defined as a sum (a Fourier series) of all possible wave states, where the coefficients of this sum are the creation and annihilation operators. This makes the field itself an operator.

• (42:24) It's explained that while the a†(k) operator creates a particle with a definite momentum, the field operator ψ†(x) creates a particle at a definite position.

• (1:05:37) The lecture describes how this framework leads to stimulated emission. The probability of creating a particle in a certain state is increased if there are already other identical particles (bosons) in that same state. This is a statistical effect due to the nature of indistinguishable particles.

• (1:18:59) A distinction is made between bosons, which tend to occupy the same state, and fermions, which are governed by the Pauli exclusion principle and cannot be in the same state.

• (1:40:43) The lecture concludes by explaining that a quantum field behaves like a classical field when the number of quanta (particles) is very large. The discreteness of particles only becomes apparent when the number of quanta is small.

In this lecture Professor Susskind continues on the subject of quantum field theory.

Summary: 

• (03:24) The lecture begins by explaining the Dirac Delta function, a mathematical tool used to represent a quantity that is infinitely concentrated at a single point, like an idealized point charge or mass. It is defined as a function that is zero everywhere except at a single point, where it is infinite, yet its integral over its entire domain is equal to one.

• (15:41) The concepts of bra-ket notation are introduced as a way to represent quantum states. A "ket" |ψ⟩ represents the state of a quantum system, while a "bra" ⟨φ| represents a linear functional that acts on kets. The combination ⟨φ|ψ⟩, a "bra-ket," represents the inner product between two states, which gives the probability amplitude for the system in state |ψ⟩ to be found in state |φ⟩.

• (19:24) Creation and annihilation operators are discussed as operators that, respectively, increase or decrease the number of particles in a given quantum state. The lecture details the rules for how these operators act on both bra and ket vectors, showing that their actions are interchanged when moving from one to the other.

• (33:31) The lecture defines a quantum field as an operator-valued function of space and time. It is constructed from a superposition of creation and annihilation operators for each possible momentum, effectively associating an operator with every point in spacetime.

• (40:19) It is shown that for a non-relativistic particle, the defined quantum field satisfies the Schrödinger equation. This connects the abstract quantum field theory back to the familiar wave equation of elementary quantum mechanics.

• (53:02) A simple model of a scattering process is presented, where a particle interacts with a fixed target. This interaction is described using creation and annihilation operators: the incoming particle is annihilated at the target's location, and an outgoing particle is created.

• (1:03:31) The principle of energy conservation is shown to be a direct consequence of the assumption that the scattering interaction can happen at any time. Integrating over all possible interaction times mathematically forces the initial and final energies to be equal.

• (1:35:37) Similarly, charge conservation is linked to "phase invariance." The interactions described must be unchanged by a change in the quantum field's phase. This condition requires an equal number of creation and annihilation operators in the interaction term, which corresponds to conserving the number of particles (and thus charge).

In this lecture Professor Susskind continues on the subject of quantum field theory, more specifically, energy conservation, waves and fermions.

Summary:

• Phase vs. Group Velocity The lecture begins by distinguishing between phase velocity (the speed of a point of constant phase on a wave) and group velocity (the speed of the overall shape of the wave's amplitude). For a plane wave, the phase velocity is given by ω/k. However, in quantum mechanics, the physically meaningful velocity that describes the motion of particles and signals is the group velocity, given by the derivative dω/dk.

• Schrödinger Waves For a non-relativistic particle described by the Schrödinger equation, the energy (ω) is proportional to the momentum (k) squared (ω = k²/2m). This leads to a phase velocity of k/2m, which is half the classical velocity (p/m). The group velocity, however, correctly gives the classical velocity k/m. This illustrates that phase velocity can be an artifact of mathematical conventions and is not always physically measurable.

• Relativistic Waves For a relativistic particle, the relationship between energy and momentum is ω = √(k² + m²). In this case, the phase velocity (ω/k) is always greater than the speed of light, while the group velocity (dω/dk) is always less than the speed of light. The product of the group and phase velocities is equal to the speed of light squared (in this case, 1).

• Symmetries and Conservation Laws The lecture explains the deep connection between symmetries and conservation laws in quantum field theory. Time translation invariance leads to the conservation of energy, while spatial translation invariance leads to the conservation of momentum. Symmetries related to the phase of the wave function are associated with conservation laws like the conservation of electric charge.

• Bosons and Fermions The lecture then contrasts the two fundamental types of particles: bosons and fermions.

  • Bosons can occupy the same quantum state, and in the ground state, a collection of bosons will all condense into the lowest possible energy state (0 momentum).

  • Fermions, on the other hand, obey the Pauli exclusion principle, which states that no two identical fermions can occupy the same quantum state. This leads to the concept of the Fermi sphere, where in the ground state, fermions fill up all available energy levels from the lowest energy up to a certain maximum energy called the Fermi energy.

• Creation and Annihilation Operators The mathematical formalism for handling these particles involves creation and annihilation operators. For fermions, these operators follow anti-commutation relations, which mathematically enforce the exclusion principle. For example, applying the creation operator twice for the same state results in zero, meaning it's impossible to have two fermions in the same state.

• Holes and Antiparticles Excitations in a Fermi system involve moving a fermion from a state within the Fermi sphere to a state outside of it. This creates an electron (the excited particle) and a hole (the empty state left behind). This hole behaves like a particle with the opposite charge of the original particle and is an early concept related to the idea of antiparticles.

• The Dirac Equation The lecture concludes with a simplified, one-dimensional version of the Dirac equation, which describes relativistic fermions. This equation famously predicted the existence of negative energy states. Dirac's ingenious solution was to propose that the "vacuum" is actually a "sea" where all negative energy states are already filled by fermions (the "Dirac sea"). A hole in this sea would then be observed as a particle with the same mass but opposite charge—an antiparticle, which was later discovered to be the positron.

In this lecture Professor Susskind continues on the subject of quantum field theory, including, the Dirac equation and Higgs particles.

Summary:

• The Dirac Equation and Antiparticles: The lecture starts with a simple version of the Dirac equation, which describes particles like electrons. This equation naturally leads to solutions with both positive and negative energy. To solve the problem of particles endlessly falling to lower energy states, Dirac proposed that all negative energy states are already filled. A hole in this "sea" of negative energy particles behaves as an antiparticle with a positive charge and positive energy.

• Right-Movers and Left-Movers: In one dimension, the simplest equation only describes particles moving to the right. To make the theory symmetric, it's expanded to include two types of particles: "right-movers" and "left-movers." The vacuum is envisioned as a balanced sea filled with both types of negative-energy particles.

• How Particles Acquire Mass: Massless particles travel at the speed of light. In this framework, a particle acquires mass when the equations for its left-moving and right-moving components are coupled together. This coupling mixes the two, and a massive particle at rest is understood as a quantum superposition of its left- and right-moving parts.

• Matrix Representation: The coupled equations for massive particles are elegantly expressed using matrices, named alpha (α) and beta (β). These matrices have specific properties (for example, they "anti-commute," meaning αβ + βα = 0) that are crucial for correctly deriving the relativistic energy-momentum relationship (ω² = k² + m²).

• From One Dimension to Three: The concept is then expanded from one to three spatial dimensions. This generalization requires upgrading the 2x2 matrices to larger 4x4 matrices. This more complex structure naturally gives rise to an intrinsic property of particles that was not present in the 1D model: spin.

• The Higgs Field and Mass: The lecture connects this mathematical description of mass to the Higgs mechanism. The mass term in the Dirac equation can be viewed as the result of a particle's interaction with the Higgs field, which has a non-zero value throughout the vacuum. Particles that we observe to have mass are fundamentally massless but acquire their effective mass from this interaction. The Higgs boson itself is an excitation, or vibration, of this field.

Leonard Susskind discusses the theory and mathematics of angular momentum.

Summary:

• Angular Momentum: This is a vector quantity that describes an object's rotation. Its direction is along the axis of rotation, determined by the right-hand rule. It comes in two forms: orbital angular momentum (from the motion of its center of mass) and spin angular momentum (the intrinsic rotation of a particle in its own rest frame).

• Quantum Spin: In quantum mechanics, elementary particles have a fixed, intrinsic spin that characterizes them. Unlike a basketball, you cannot simply "spin up" an electron. The energy required to change its intrinsic spin is so immense it would effectively turn it into a different particle.

• Quantization: A key principle is that angular momentum is quantized, meaning it only exists in discrete units. For a given angular momentum, a smaller object requires more energy to spin. For a particle as small as an electron, the energy needed to give it one more unit of angular momentum is astronomical.

• Measurement Limitations: The different components of angular momentum (e.g., along the x, y, and z axes) do not commute in quantum mechanics. This means you can only precisely measure one component at a time. Typically, the z-axis is chosen as the "quantization axis".

• Spectrum of Spin: Through the use of "raising" and "lowering" operators, it's shown that the possible values for the measured spin component are spaced by integers. This leads to two families of particles: those with integer spin (0, 1, 2...) and those with half-integer spin (1/2, 3/2, 5/2...).

• Fermions and Bosons: This spin value is fundamentally linked to a particle's nature.

  • Half-integer spin particles are fermions (like electrons and quarks). They obey the Pauli Exclusion Principle, meaning no two can occupy the same quantum state.

  • Integer spin particles are bosons (like photons). They do not have this restriction and can pile into the same state.

• Pauli Exclusion Principle: This principle, crucial for chemistry, states that you cannot put more than one electron into the same total quantum state. This is why atomic orbitals can hold a maximum of two electrons, as long as they have opposite spins.

• Composite Particles: The nature of a composite particle depends on its constituents. Combining an even number of fermions creates a boson (e.g., a hydrogen atom), while an odd number results in a fermion.

Leonard Susskind discusses the theory and mathematics of particle spin and half spin, the Dirac equation, and isotopic spin.

Summary: 

• (00:01) The lecture introduces the core distinction between bosons and fermions through the symmetry of their wave functions. Bosonic wave functions are symmetric upon the interchange of particles, which allows multiple bosons to occupy the same quantum state. In contrast, fermionic wave functions are anti-symmetric, meaning they change sign, which leads to the Pauli exclusion principle where no two fermions can be in the same quantum state.

• (03:56) A method for constructing appropriate wave functions for bosons and fermions is demonstrated. A symmetric wave function for bosons can be created by adding a wave function to its interchanged-particle counterpart. An anti-symmetric wave function for fermions is created by subtracting the interchanged version.

• (05:25) Using the hydrogen atom as an example, the lecture explains how a composite particle made of fermions (an electron and a proton) can behave as a boson. While the overall wave function is anti-symmetric with respect to the interchange of the individual electrons or protons, it is symmetric when two entire hydrogen atoms are interchanged.

• (21:40) The discussion shifts to the mathematics of spin, highlighting its foundational role in understanding other particle physics concepts like isospin and color. The commutation relations that govern spin angular momentum are presented.

• (26:00) For spin-1/2 particles like electrons, their state is described by a two-component vector known as a spinner. The Pauli spin matrices are introduced as the 2x2 matrix representation of the spin operators, and the lecture explains how to determine the eigenvectors for spin up and down along different spatial axes.

• (45:30) In the case of spin-1 particles, there are three possible spin states (+1, 0, -1), resulting in a three-dimensional state space. The lecture presents a 3x3 matrix representation for the spin operators and their corresponding eigenvectors.

• (55:15) The concept of a wave function is expanded to include spin for particles moving in space. For a spin-1/2 particle, the wave function has two components (spin up and down) that are functions of position, while a spin-1 particle's wave function has three such components.

• (1:08:40) A connection is made between spin and the Dirac equation. The 4x4 matrices used in this equation are explained, and the four components of the Dirac spinner are shown to correspond to the two spin states for both positive and negative energy solutions, which represent particles and anti-particles.

• (1:32:55) The negative energy solutions of the Dirac equation lead to the theoretical prediction of anti-particles, like the positron. The "Dirac sea" model is introduced to explain this, where the vacuum is envisioned as being filled with negative-energy electrons. A hole in this sea behaves like a particle with positive energy and opposite charge.

Leonard Susskind discusses the equations of motion of fields containing particles and quantum field theory, and shows how basic processes are coded by a Lagrangian.

Summary: 

• (00:50) In modern physics, the equations of motion for all particles and fields are derived from a single, compact expression called a Lagrangian. It's a foundational concept that contains all the dynamic information of a system.

• (02:48) A Lagrangian is a function of a field and its derivatives. By applying a specific set of rules (the Euler-Lagrange equations) to this function, one can generate the wave equations that describe the behavior of the corresponding particles.

• (12:20) For a simple, non-interacting particle, the Lagrangian leads to the famous energy-momentum relation (E² = p²c² + m²c⁴), where the parameter 'm' in the Lagrangian is identified as the mass of the particle.

• (15:14) Terms in the Lagrangian that are of a higher power than quadratic (e.g., cubic or quartic) are called nonlinear terms. These terms represent interactions, causing particles to scatter, decay, or annihilate each other rather than passing through one another unchanged.

• (22:02) A Lagrangian can include multiple fields. When a term in the Lagrangian contains more than one type of field, it signifies an interaction between them. This results in coupled equations of motion where the presence of one particle field affects the behavior of another.

• (29:20) In quantum field theory, the fields are composed of creation and annihilation operators. A term like φ³ in a Lagrangian represents physical processes such as a single particle decaying into two, two particles combining into one, or the creation of three particles from nothing.

• (32:36) The quadratic terms in the Lagrangian govern the free motion of a particle, describing how it moves from one point in spacetime to an adjacent one. In contrast, the higher-order interaction terms describe events happening at a single point.

• (41:37) The fundamental interaction in quantum electrodynamics (QED) is described by a single term in the Lagrangian that couples the electron field (ψ), the positron field, and the photon field (A). This one term accounts for all electromagnetic processes, such as an electron emitting or absorbing a photon, or an electron and a positron annihilating into a photon.

• (1:13:59) A core principle of quantum field theory is locality. All fundamental processes described in the Lagrangian occur at a single point in spacetime. Complex, non-local events, like a photon traveling from a star to your eye, are built up from a series of these elementary local interactions.

• (1:34:17) Symmetries in the Lagrangian lead directly to conservation laws. For example, a specific phase symmetry in the QED Lagrangian ensures the conservation of electric charge. Physicists use these symmetries to guess the form of the Lagrangian that describes observed particle interactions.

In this lecture, Professor Susskind elaborates further on using field Lagrangians, the action principle, and path integrals in studying particle physics.

Summary: 

• Path Integral Method: Quantum field theory utilizes the path integral method, which is the quantum mechanical equivalent of the principle of least action, to describe particle interactions.

• Action and Lagrangian: The action is the integral of the Lagrangian along a trajectory. In classical physics, a particle follows the path of least action. This principle is extended to fields in spacetime in quantum field theory.

• Quantum Amplitudes: The amplitude for a particle to travel between two points is calculated by summing e^(-i * Action) over every possible path, not just the classical one. The probability is the square of this amplitude.

• Fields to Particles: The path integral method can also be used for fields to determine the probability of a field configuration changing over time. This concept can be rephrased in terms of particles, which are the quanta of the field.

• Feynman Diagrams: Particle interactions are visually represented by Feynman diagrams, which illustrate particles moving, splitting, and merging. The principles for these diagrams are based on the theory's Lagrangian.

• Lagrangian Components: Various terms in the Lagrangian correspond to different physical events. Kinetic terms describe the propagation of particles, mass terms influence the paths, and interaction terms describe the creation and annihilation of particles.

• Calculating Probabilities: The amplitude of a complex process is the sum of the amplitudes of all possible Feynman diagrams representing it. Diagrams with more vertices (interactions) generally have a smaller contribution to the total amplitude, provided the coupling constants are small.

• Conservation Laws: The laws of conservation of energy and momentum are natural outcomes of the calculations when integrating over all possible spacetime positions of the interactions.

• The Standard Model: The lecture concludes by preparing to discuss the Standard Model of particle physics, which catalogues all known elementary particles and their interactions as described by a Lagrangian.