120

Nice discovery! The formula for time dilation outside a spherical body is $$\tau = t\sqrt{1-\frac{2GM}{c^2r}}$$ where $\tau$ is the proper time as measured by your object at coordinate radius $r$, $t$ is the time as measured by an observer at infinity, $M$ the mass of the spherical body, and $G$ and $c$ the gravitational constant and the speed of light. ...


52

TL;DR: Yes, it is just a short-cut. The main point is that the complexified map $$\tag{A} \begin{pmatrix} \phi \\ \phi^{*} \end{pmatrix} ~=~ \begin{pmatrix} 1 & i\\ 1 &-i \end{pmatrix} \begin{pmatrix} \phi_1 \\ \phi_2 \end{pmatrix} $$ is a bijective map :$\mathbb{C}^2 \to \mathbb{C}^2 $. Notation in this answer: In this answer, let $\phi,\phi^{*}...


43

The nature of complex numbers in QM turned up in a recent discussion, and I got called a stupid hack for questioning their relevance. Mainly for therapeutic reasons, I wrote up my take on the issue: On the Role of Complex Numbers in Quantum Mechanics Motivation It has been claimed that one of the defining characteristics that separate the quantum world ...


36

I) Well, one can identify a complex-valued observable with a normal operator $$\tag{1} A^{\dagger}A~=~AA^{\dagger}.$$ A version$^1$ of the spectral theorem states that an operator $A$ is orthonormally diagonalizable iff $A$ is a normal operator. Thus normal operators are the only kind of operators that we can consistently extract measurements [i.e. ...


36

A scalar is a one-component quantity that is invariant under rotations of the coordinate system OK, but what then do you mean by "rotation"? See, a scalar in the sense as defined in your quote is not just "a scalar", period. You can only have a scalar with respect to some particular rotation operation. The same quantity can be a scalar with respect to one ...


34

There are some problems with using quaternions to describe spacetime. Quaternions have two important properties: (1) they form a four-dimensional vector space; (2) you can multiply quaternions together.[1] The first property is obviously very suggestive, but it's no different from the usual four-vectors that we already use in special relativity. To ...


31

Complex numbers are used in all of mathematics, and therefore by extension they are used in other fields that require math; not just physics, but also engineering and other fields. Trying to assign a "physical interpretation" to a complex number would be like assigning a physical interpretation to a real number, such as the number 5. A complex number is ...


30

A simple reference problem Suppose we want to analyse the problem of a forced harmonic oscillator. Denote as $\phi(t)$ the time dependent position of the oscillator. The oscillator experiences two forces, the spring force $-k\phi(t)$ and an external force $F_{\text{ext}}(t)$. Newton's law says $$ \begin{align} F(t) &= m a(t) \\ -k \phi(t) + F_{\text{...


28

The physical 'meaning' of the imaginary part of the impedance is that it represents the energy storage part of the circuit element. To see this, let the sinusoidal current $i = I\cos(\omega t)$ be the current through a series RL circuit. The voltage across the combination is $$v = Ri + L\frac{di}{dt} = RI\cos(\omega t) - \omega LI\sin(\omega t)$$ The ...


27

Complex numbers are used in "macro" physics. They are used in analysis of electrical circuits (especially when AC is involved) and in fluid dynamics. Solution of differential equations is simplified if complex numbers are used, as is Fourier analysis. Any scenario that involves periodic or cyclic functions can be modeled using complex numbers.


24

This year-old question popped up unexpectedly when I signed in, and it's an interesting one. So I guess it's OK just to add an intuition-level "addendum answer" to the excellent and far more complete responses provided long ago. Your kernel question seems to be this: "Why is the wave function complex?" My intentionally informal answer is this: Because ...


23

The easiest way to see imaginary time used is in elementary quantum mechanics in one dimension. (This is the explanation cribbed from wikipedia). Suppose we're looking at a tunneling-through-a-barrier problem. We start with the Schrodinger equation: $$ -\frac{\hbar^2}{2m}\frac{d^2\psi(x)}{dx^2}+V(x)\psi(x) = E\psi(x) $$ Make the ansatz $$ \psi(x) ...


23

Short answer: complexifications facilitate representation theory. In physics, we typically want to find representations of a Lie algebra $\mathfrak g$, and often times determining the representations of its complexification $\mathfrak g_\mathbb C$ is easier. Moreover, we have the following theorem (see ref 1. Proposition 4.6) which tells us that ...


21

as you wrote, the spacetime invariant can be expressed as: $$ds^2=g_{\mu\nu}dx^{\mu}dx^{\nu}$$ and from that we normally get: $$ds^2=-c^2dt^2+dx^2+dy^2+dz^2$$ This is not because of some arbitrary imaginary time unit, this is because the metric ($g_{\mu\nu}$) is a diagonal matrix with the coefficients of each term of the $ds^2$ equation: $$g_{\mu\nu}=\left(\...


21

The basis vectors are exactly what you'd expect, $$(1, 0, 0, 0), \quad (0, 1, 0, 0), \quad (0, 0, 1, 0), \quad (0, 0, 0, 1).$$ However, the inner product, i.e. the way we combine two vectors into a number, is not the same as the usual dot product. Using your notation, we're changing the definition of $\cdot$, not the definition of the $e_{\mu}$. You are ...


18

If you don't like complex numbers, you can use pairs of real numbers (x,y). You can "add" two pairs by (x,y)+(z,w) = (x+z,y+w), and you can "multiply" two pairs by (x,y) * (z,w) = (xz-yw, xw+yz). (If don't think that multiplication should work that way, you can call this operation "shmultiplication" instead.) Now you can do anything in quantum mechanics. ...


18

First of all, why are Fourier transforms useful? The Fourier transform is special because the complex exponential functions are eigenfunctions of he translation. In any linear problem with translation invariance, the Fourier transform turns a differential equation into an algebraic one. A simple example is a driven damped harmonic oscillator. The equation ...


18

It doesn't really play a role (in a way), or at least not as far as physical results go. Whenever someone says we consider a plane wave of the form $f(x) = Ae^{i(kx-\omega t)}$, what they are really saying is something like we consider an oscillatory function of the form $f_\mathrm{re}(x) = |A|\cos(kx-\omega t +\varphi)$, but: we can represent ...


17

We don't have to modify the basic laws of quantum mechanics to describe unstable particles. The full state of the system is includes the state of the decay products, and what you really have is a coupling from one state to another. No imaginary energies are required to describe this, but you do need to include the states of the decay products in your ...


17

As a mathematical structure, the field of complex numbers does not admit an order relation which is an extension of the order we have in $\mathbb{R}$. This means that there is absolutely no way of saying if $5+3i$ is bigger or smaller than $5+6i$ for example. We just know it is not equal and we have to stop here. Therefore it is physically really hard (...


17

The significance of the metric: $$ d\tau^2 = dt^2 - dx^2 $$ is that $d\tau^2$ is an invarient i.e. every observer in every frame, even accelerated frames, will agree on the value of $d\tau^2$. In contrast $dt$ and $dx$ are coordinate dependant and different observers will disagree about the relative values of $dt$ and $dx$. So while it is certainly true ...


17

You can easily eliminate all references to complex numbers in a rather trivial way, although doing so results in much less mathematically elegant expressions. For example, you could choose to work in the position basis and, instead of using a complex-valued wavefunction that assigns a complex number to every point in configuration space, you could use a ...


15

Frank, I would suggest buying or borrowing a copy of Richard Feynman's QED: The Strange Theory of Light and Matter. Or, you can just go directly to the online New Zealand video version of the lectures that gave rise to the book. In QED you will see how Feynman dispenses with complex numbers entirely, and instead describes the wave functions of photons (...


14

Complex analysis is more than just a tool that can be used for computing difficult integrals. For example: In quantum field theory, one of the most popular regularization schemes relies on the theory of complex functions. In particular, it relies on the concept of analytic continuation of functions $f:D\to\mathbb C$ for some $D\subseteq \mathbb C$. In ...


14

You may indeed identify the generators in the way you did. However, the Lie algebras and Lie groups are different because – as quickly said by Qmechanic – you must use different reality conditions for the coefficients. A general matrix in the $SU(2)$ group is written as $$ M = \exp[ i( \alpha J_+ + \bar\alpha J_- + \gamma J_0 )] $$ where $\alpha\in {\...


13

Let the old master Dirac speak: "One might think one could measure a complex dynamical variable by measuring separately its real and pure imaginary parts. But this would involve two measurements or two observations, which would be alright in classical mechanics, but would not do in quantum mechanics, where two observations in general interfere ...


13

There is a fundamental result already conjectured by von Neumann but proved just at the end 20'th century by Solèr (in addition to a partial result already obtained by Piron in the sixties) which establishes (relying on the theory of orthomodular lattices and projective geometry) that the general phenomenology of Quantum Mechanics can be described only by ...


13

but why isn't SR taught with an imaginary time coordinate as standard? From "Gravitation", page 51, via Google books. I'll type up a paraphrase later.


13

The ladder operators do belong to the real Lie algebras$^1$ $$\begin{align} su(1,1)&:=~\{m\in {\rm Mat}_{2\times 2}(\mathbb{C}) \mid m^{\dagger}\sigma_3=-\sigma_3m,~ {\rm tr}(m)=0\} ~=~{\rm span}_{\mathbb{R}}\{ \sigma_1, \sigma_2, i\sigma_3 \}\cr ~\cong~sl(2,\mathbb{R}) &:=~\{m\in {\rm Mat}_{2\times 2}(\mathbb{R}) \mid {\rm tr}(m)=0\} ~=~{\rm span}...


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